on approximate birkhoff orthogonality in normed...
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On approximate Birkhoff orthogonality in normed spaces
Jacek Chmielinski
Instytut MatematykiUniwersytet Pedagogiczny w Krakowie
Banach Spaces and their ApplicationsLviv (Ukraine), June 26-29, 2019
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 1 / 28
Introduction — inner product space
(X , 〈·|·〉) — inner product space; x⊥y ⇔ 〈x |y〉 = 0.
Approximate orthogonality (ε-orthogonality with ε ∈ [0, 1)):
x⊥ε y ⇔ | 〈x |y〉 | ≤ ε ‖x‖ ‖y‖, x , y ∈ X .
Observation
x⊥ε y ⇔ ∃ z ∈ X : x⊥z , ‖z − y‖ ≤ ε‖y‖.
Indeed, if x⊥ε y take z = − 〈x |y〉‖x‖2 x + y (z = y in case x = 0).
Conversely, assuming x⊥z and ‖z − y‖ ≤ ε‖y‖,
| 〈x |y〉 | = | 〈x |y − z〉 | ≤ ‖x‖ ‖y − z‖ ≤ ε‖x‖ ‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 2 / 28
Introduction — inner product space
(X , 〈·|·〉) — inner product space; x⊥y ⇔ 〈x |y〉 = 0.
Approximate orthogonality (ε-orthogonality with ε ∈ [0, 1)):
x⊥ε y ⇔ | 〈x |y〉 | ≤ ε ‖x‖ ‖y‖, x , y ∈ X .
Observation
x⊥ε y ⇔ ∃ z ∈ X : x⊥z , ‖z − y‖ ≤ ε‖y‖.
Indeed, if x⊥ε y take z = − 〈x |y〉‖x‖2 x + y (z = y in case x = 0).
Conversely, assuming x⊥z and ‖z − y‖ ≤ ε‖y‖,
| 〈x |y〉 | = | 〈x |y − z〉 | ≤ ‖x‖ ‖y − z‖ ≤ ε‖x‖ ‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 2 / 28
Introduction — inner product space
(X , 〈·|·〉) — inner product space; x⊥y ⇔ 〈x |y〉 = 0.
Approximate orthogonality (ε-orthogonality with ε ∈ [0, 1)):
x⊥ε y ⇔ | 〈x |y〉 | ≤ ε ‖x‖ ‖y‖, x , y ∈ X .
Observation
x⊥ε y ⇔ ∃ z ∈ X : x⊥z , ‖z − y‖ ≤ ε‖y‖.
Indeed, if x⊥ε y take z = − 〈x |y〉‖x‖2 x + y (z = y in case x = 0).
Conversely, assuming x⊥z and ‖z − y‖ ≤ ε‖y‖,
| 〈x |y〉 | = | 〈x |y − z〉 | ≤ ‖x‖ ‖y − z‖ ≤ ε‖x‖ ‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 2 / 28
Introduction — inner product space
(X , 〈·|·〉) — inner product space; x⊥y ⇔ 〈x |y〉 = 0.
Approximate orthogonality (ε-orthogonality with ε ∈ [0, 1)):
x⊥ε y ⇔ | 〈x |y〉 | ≤ ε ‖x‖ ‖y‖, x , y ∈ X .
Observation
x⊥ε y ⇔ ∃ z ∈ X : x⊥z , ‖z − y‖ ≤ ε‖y‖.
Indeed, if x⊥ε y take z = − 〈x |y〉‖x‖2 x + y (z = y in case x = 0).
Conversely, assuming x⊥z and ‖z − y‖ ≤ ε‖y‖,
| 〈x |y〉 | = | 〈x |y − z〉 | ≤ ‖x‖ ‖y − z‖ ≤ ε‖x‖ ‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 2 / 28
Introduction — inner product space
(X , 〈·|·〉) — inner product space; x⊥y ⇔ 〈x |y〉 = 0.
Approximate orthogonality (ε-orthogonality with ε ∈ [0, 1)):
x⊥ε y ⇔ | 〈x |y〉 | ≤ ε ‖x‖ ‖y‖, x , y ∈ X .
Observation
x⊥ε y ⇔ ∃ z ∈ X : x⊥z , ‖z − y‖ ≤ ε‖y‖.
Indeed, if x⊥ε y take z = − 〈x |y〉‖x‖2 x + y (z = y in case x = 0).
Conversely, assuming x⊥z and ‖z − y‖ ≤ ε‖y‖,
| 〈x |y〉 | = | 〈x |y − z〉 | ≤ ‖x‖ ‖y − z‖ ≤ ε‖x‖ ‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 2 / 28
Introduction — inner product space
(X , 〈·|·〉) — inner product space; x⊥y ⇔ 〈x |y〉 = 0.
Approximate orthogonality (ε-orthogonality with ε ∈ [0, 1)):
x⊥ε y ⇔ | 〈x |y〉 | ≤ ε ‖x‖ ‖y‖, x , y ∈ X .
Observation
x⊥ε y ⇔ ∃ z ∈ X : x⊥z , ‖z − y‖ ≤ ε‖y‖.
Indeed, if x⊥ε y take z = − 〈x |y〉‖x‖2 x + y (z = y in case x = 0).
Conversely, assuming x⊥z and ‖z − y‖ ≤ ε‖y‖,
| 〈x |y〉 | = | 〈x |y − z〉 | ≤ ‖x‖ ‖y − z‖ ≤ ε‖x‖ ‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 2 / 28
Birkhoff orthogonality
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 3 / 28
Birkhoff orthogonalityG. Birkhoff, Orthogonality in linear metric spaces. Duke Math. J., 1 (1935), 169–172.
(X , ‖ · ‖) a real normed space.
x⊥By :⇐⇒ ∀λ ∈ R : ‖x + λy‖ ≥ ‖x‖.
xy
x+λy
Figure: R2 with the maximum norm; x⊥By
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 4 / 28
Birkhoff orthogonalityG. Birkhoff, Orthogonality in linear metric spaces. Duke Math. J., 1 (1935), 169–172.
(X , ‖ · ‖) a real normed space.
x⊥By :⇐⇒ ∀λ ∈ R : ‖x + λy‖ ≥ ‖x‖.
xy
x+λy
Figure: R2 with the maximum norm; x⊥By
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 4 / 28
Birkhoff orthogonalityG. Birkhoff, Orthogonality in linear metric spaces. Duke Math. J., 1 (1935), 169–172.
(X , ‖ · ‖) a real normed space.
x⊥By :⇐⇒ ∀λ ∈ R : ‖x + λy‖ ≥ ‖x‖.
xy
x+λy
Figure: R2 with the maximum norm; x⊥By
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 4 / 28
Birkhoff orthogonalityG. Birkhoff, Orthogonality in linear metric spaces. Duke Math. J., 1 (1935), 169–172.
(X , ‖ · ‖) a real normed space.
x⊥By :⇐⇒ ∀λ ∈ R : ‖x + λy‖ ≥ ‖x‖.
xy
x+λy
Figure: R2 with the maximum norm; x⊥By
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 4 / 28
Approximate Birkhoff orthogonality
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 5 / 28
Approximate Birkhoff orthogonality
For ε ∈ [0, 1) we consider an ε-Birkhoff orthogonality ⊥εB.
J. Chmielinski, On an ε-Birkhoff orthogonality, J. Inequal. Pure andAppl. Math. 6 (2005), Art. 79.
x⊥εBy :⇐⇒ ∀λ ∈ K : ‖x + λy‖2 ≥ ‖x‖2 − 2ε‖x‖ ‖λy‖.
J. Chmielinski, T. Stypu la, P. Wojcik, Approximate orthogonality innormed spaces and its applications, Linear Algebra and itsApplications 531 (2017), 305–317.
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 6 / 28
Approximate Birkhoff orthogonality
For ε ∈ [0, 1) we consider an ε-Birkhoff orthogonality ⊥εB.
J. Chmielinski, On an ε-Birkhoff orthogonality, J. Inequal. Pure andAppl. Math. 6 (2005), Art. 79.
x⊥εBy :⇐⇒ ∀λ ∈ K : ‖x + λy‖2 ≥ ‖x‖2 − 2ε‖x‖ ‖λy‖.
J. Chmielinski, T. Stypu la, P. Wojcik, Approximate orthogonality innormed spaces and its applications, Linear Algebra and itsApplications 531 (2017), 305–317.
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 6 / 28
Approximate Birkhoff orthogonality
For ε ∈ [0, 1) we consider an ε-Birkhoff orthogonality ⊥εB.
J. Chmielinski, On an ε-Birkhoff orthogonality, J. Inequal. Pure andAppl. Math. 6 (2005), Art. 79.
x⊥εBy :⇐⇒ ∀λ ∈ K : ‖x + λy‖2 ≥ ‖x‖2 − 2ε‖x‖ ‖λy‖.
J. Chmielinski, T. Stypu la, P. Wojcik, Approximate orthogonality innormed spaces and its applications, Linear Algebra and itsApplications 531 (2017), 305–317.
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 6 / 28
Approximate Birkhoff orthogonality
For ε ∈ [0, 1) we consider an ε-Birkhoff orthogonality ⊥εB.
J. Chmielinski, On an ε-Birkhoff orthogonality, J. Inequal. Pure andAppl. Math. 6 (2005), Art. 79.
x⊥εBy :⇐⇒ ∀λ ∈ K : ‖x + λy‖2 ≥ ‖x‖2 − 2ε‖x‖ ‖λy‖.
J. Chmielinski, T. Stypu la, P. Wojcik, Approximate orthogonality innormed spaces and its applications, Linear Algebra and itsApplications 531 (2017), 305–317.
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 6 / 28
Approximate Birkhoff orthogonality
For ε ∈ [0, 1) we consider an ε-Birkhoff orthogonality ⊥εB.
J. Chmielinski, On an ε-Birkhoff orthogonality, J. Inequal. Pure andAppl. Math. 6 (2005), Art. 79.
x⊥εBy :⇐⇒ ∀λ ∈ K : ‖x + λy‖2 ≥ ‖x‖2 − 2ε‖x‖ ‖λy‖.
J. Chmielinski, T. Stypu la, P. Wojcik, Approximate orthogonality innormed spaces and its applications, Linear Algebra and itsApplications 531 (2017), 305–317.
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 6 / 28
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
1−1
1
−1
x
y
z
Figure: R2 with l∞-l1-norm
x 6⊥By , x⊥z , ‖z − y‖ ≤ ε⇒ x⊥εBy .
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 7 / 28
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
1−1
1
−1
x
y
z
Figure: R2 with l∞-l1-norm
x 6⊥By , x⊥z , ‖z − y‖ ≤ ε⇒ x⊥εBy .
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 7 / 28
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
1−1
1
−1
x
y
z
Figure: R2 with l∞-l1-norm
x 6⊥By
, x⊥z , ‖z − y‖ ≤ ε⇒ x⊥εBy .
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 7 / 28
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
1−1
1
−1
x
y
z
Figure: R2 with l∞-l1-norm
x 6⊥By , x⊥z
, ‖z − y‖ ≤ ε⇒ x⊥εBy .
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 7 / 28
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
1−1
1
−1
x
y
z
Figure: R2 with l∞-l1-norm
x 6⊥By , x⊥z , ‖z − y‖ ≤ ε
⇒ x⊥εBy .
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 7 / 28
x⊥εBy ⇐⇒ ∃z ∈ Lin{x , y} : x⊥Bz , ‖z − y‖ ≤ ε‖y‖.
1−1
1
−1
x
y
z
Figure: R2 with l∞-l1-norm
x 6⊥By , x⊥z , ‖z − y‖ ≤ ε⇒ x⊥εBy .
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 7 / 28
For 0 6= x ∈ X we consider the class of its supporting functionals:
J(x) = {ϕ ∈ X ∗ : ‖ϕ‖ = 1, ϕ(x) = ‖x‖ }.
Theorem
Let X be a real normed space, x , y ∈ X and ε ∈ [0, 1). Then
x⊥εBy ⇔ ∃ϕ ∈ J(x) : |ϕ(y)| ≤ ε‖y‖.
In particular (James),
x⊥By ⇔ ∃ϕ ∈ J(x) : ϕ(y) = 0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 8 / 28
For 0 6= x ∈ X we consider the class of its supporting functionals:
J(x) = {ϕ ∈ X ∗ : ‖ϕ‖ = 1, ϕ(x) = ‖x‖ }.
Theorem
Let X be a real normed space, x , y ∈ X and ε ∈ [0, 1). Then
x⊥εBy ⇔ ∃ϕ ∈ J(x) : |ϕ(y)| ≤ ε‖y‖.
In particular (James),
x⊥By ⇔ ∃ϕ ∈ J(x) : ϕ(y) = 0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 8 / 28
For 0 6= x ∈ X we consider the class of its supporting functionals:
J(x) = {ϕ ∈ X ∗ : ‖ϕ‖ = 1, ϕ(x) = ‖x‖ }.
Theorem
Let X be a real normed space, x , y ∈ X and ε ∈ [0, 1). Then
x⊥εBy ⇔ ∃ϕ ∈ J(x) : |ϕ(y)| ≤ ε‖y‖.
In particular (James),
x⊥By ⇔ ∃ϕ ∈ J(x) : ϕ(y) = 0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 8 / 28
Applications
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 9 / 28
Orthogonality of operators on a Hilbert space
H – Hilbert space; L(H) – the space of linear bounded operators on H.For T ∈ L(H):
MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
R. Bhatia, P. Semrl, Orthogonality of matrices and some distanceproblems, Linear Algebra Appl. 287 (1999), 77-85.
Theorem (Bhatia-Semrl)
Let H be a Hilbert space and let T ,S ∈ L(H). Then, the followingconditions are equivalent:
(1) T⊥BS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, 〈Txn|Sxn〉 → 0 (n→∞).
Moreover, if dimH <∞ and T ,S ∈ L(H), then each of the aboveconditions is equivalent to:
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥Sx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 10 / 28
Orthogonality of operators on a Hilbert space
H – Hilbert space; L(H) – the space of linear bounded operators on H.For T ∈ L(H):
MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
R. Bhatia, P. Semrl, Orthogonality of matrices and some distanceproblems, Linear Algebra Appl. 287 (1999), 77-85.
Theorem (Bhatia-Semrl)
Let H be a Hilbert space and let T ,S ∈ L(H). Then, the followingconditions are equivalent:
(1) T⊥BS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, 〈Txn|Sxn〉 → 0 (n→∞).
Moreover, if dimH <∞ and T ,S ∈ L(H), then each of the aboveconditions is equivalent to:
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥Sx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 10 / 28
Orthogonality of operators on a Hilbert space
H – Hilbert space; L(H) – the space of linear bounded operators on H.For T ∈ L(H):
MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
R. Bhatia, P. Semrl, Orthogonality of matrices and some distanceproblems, Linear Algebra Appl. 287 (1999), 77-85.
Theorem (Bhatia-Semrl)
Let H be a Hilbert space and let T ,S ∈ L(H). Then, the followingconditions are equivalent:
(1) T⊥BS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, 〈Txn|Sxn〉 → 0 (n→∞).
Moreover, if dimH <∞ and T ,S ∈ L(H), then each of the aboveconditions is equivalent to:
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥Sx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 10 / 28
Orthogonality of operators on a Hilbert space
H – Hilbert space; L(H) – the space of linear bounded operators on H.For T ∈ L(H):
MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
R. Bhatia, P. Semrl, Orthogonality of matrices and some distanceproblems, Linear Algebra Appl. 287 (1999), 77-85.
Theorem (Bhatia-Semrl)
Let H be a Hilbert space and let T , S ∈ L(H). Then, the followingconditions are equivalent:
(1) T⊥BS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, 〈Txn|Sxn〉 → 0 (n→∞).
Moreover, if dimH <∞ and T ,S ∈ L(H), then each of the aboveconditions is equivalent to:
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥Sx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 10 / 28
Approximate orthogonality in L(H)
H – Hilbert space; T ∈ L(H); MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
Theorem
For T ,S ∈ L(H) the following conditions are equivalent:
(1) T⊥εBS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, limn→∞ | 〈Txn|Sxn〉 | ≤ ε‖T‖ ‖S‖.
Moreover, if dimH <∞, then the above conditions are equivalent to
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, | 〈Tx0|Sx0〉 | ≤ ε‖T‖ ‖S‖.
If dimH <∞ or if T is compact and if additionally MT ⊂ MS , the abovethree conditions are equivalent also to
(4) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥εSx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 11 / 28
Approximate orthogonality in L(H)
H – Hilbert space; T ∈ L(H); MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
Theorem
For T ,S ∈ L(H) the following conditions are equivalent:
(1) T⊥εBS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, limn→∞ | 〈Txn|Sxn〉 | ≤ ε‖T‖ ‖S‖.
Moreover, if dimH <∞, then the above conditions are equivalent to
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, | 〈Tx0|Sx0〉 | ≤ ε‖T‖ ‖S‖.
If dimH <∞ or if T is compact and if additionally MT ⊂ MS , the abovethree conditions are equivalent also to
(4) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥εSx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 11 / 28
Approximate orthogonality in L(H)
H – Hilbert space; T ∈ L(H); MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
Theorem
For T , S ∈ L(H) the following conditions are equivalent:
(1) T⊥εBS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, limn→∞ | 〈Txn|Sxn〉 | ≤ ε‖T‖ ‖S‖.
Moreover, if dimH <∞, then the above conditions are equivalent to
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, | 〈Tx0|Sx0〉 | ≤ ε‖T‖ ‖S‖.
If dimH <∞ or if T is compact and if additionally MT ⊂ MS , the abovethree conditions are equivalent also to
(4) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥εSx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 11 / 28
Approximate orthogonality in L(H)
H – Hilbert space; T ∈ L(H); MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
Theorem
For T , S ∈ L(H) the following conditions are equivalent:
(1) T⊥εBS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, limn→∞ | 〈Txn|Sxn〉 | ≤ ε‖T‖ ‖S‖.
Moreover, if dimH <∞, then the above conditions are equivalent to
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, | 〈Tx0|Sx0〉 | ≤ ε‖T‖ ‖S‖.
If dimH <∞ or if T is compact and if additionally MT ⊂ MS , the abovethree conditions are equivalent also to
(4) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥εSx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 11 / 28
Approximate orthogonality in L(H)
H – Hilbert space; T ∈ L(H); MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
Theorem
For T , S ∈ L(H) the following conditions are equivalent:
(1) T⊥εBS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, limn→∞ | 〈Txn|Sxn〉 | ≤ ε‖T‖ ‖S‖.
Moreover, if dimH <∞, then the above conditions are equivalent to
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, | 〈Tx0|Sx0〉 | ≤ ε‖T‖ ‖S‖.
If dimH <∞ or if T is compact and if additionally MT ⊂ MS , the abovethree conditions are equivalent also to
(4) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥εSx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 11 / 28
Approximate orthogonality in L(H)
H – Hilbert space; T ∈ L(H); MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
Theorem
For T , S ∈ L(H) the following conditions are equivalent:
(1) T⊥εBS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, limn→∞ | 〈Txn|Sxn〉 | ≤ ε‖T‖ ‖S‖.
Moreover, if dimH <∞, then the above conditions are equivalent to
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, | 〈Tx0|Sx0〉 | ≤ ε‖T‖ ‖S‖.
If dimH <∞ or if T is compact and if additionally MT ⊂ MS , the abovethree conditions are equivalent also to
(4) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥εSx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 11 / 28
Approximate orthogonality in L(H)
H – Hilbert space; T ∈ L(H); MT := {x ∈ SH : ‖Tx‖ = ‖T‖}.
Theorem
For T , S ∈ L(H) the following conditions are equivalent:
(1) T⊥εBS ;
(2) ∃ (xn)∞n=1 ⊂ SH : ‖Txn‖ → ‖T‖, limn→∞ | 〈Txn|Sxn〉 | ≤ ε‖T‖ ‖S‖.
Moreover, if dimH <∞, then the above conditions are equivalent to
(3) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, | 〈Tx0|Sx0〉 | ≤ ε‖T‖ ‖S‖.
If dimH <∞ or if T is compact and if additionally MT ⊂ MS , the abovethree conditions are equivalent also to
(4) ∃ x0 ∈ SH : ‖Tx0‖ = ‖T‖, Tx0⊥εSx0.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 11 / 28
Approximate orthogonality in C0(K )
L(H) Let K be a locally compact topological space.
C0(K ) := {f : K → R cont. : ∀ ε > 0, {t ∈ K : |f (t)| ≥ ε} compact}
– with the supremum norm. For f ∈ C0(K ), Mf := {t ∈ K : |f (t)| = ‖f ‖}(nonempty and compact).
Theorem
Let f , g ∈ C0(K ), f 6= 0 6= g . Assume that Mf is connected. Then, thefollowing conditions are equivalent:
(a) f⊥εBg ,
(b) ∃ t1 ∈ Mf : |g(t1)| ≤ ε‖g‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 12 / 28
Approximate orthogonality in C0(K )
L(H) Let K be a locally compact topological space.
C0(K ) := {f : K → R cont. : ∀ ε > 0, {t ∈ K : |f (t)| ≥ ε} compact}
– with the supremum norm. For f ∈ C0(K ), Mf := {t ∈ K : |f (t)| = ‖f ‖}(nonempty and compact).
Theorem
Let f , g ∈ C0(K ), f 6= 0 6= g . Assume that Mf is connected. Then, thefollowing conditions are equivalent:
(a) f⊥εBg ,
(b) ∃ t1 ∈ Mf : |g(t1)| ≤ ε‖g‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 12 / 28
Approximate orthogonality in C0(K )
L(H) Let K be a locally compact topological space.
C0(K ) := {f : K → R cont. : ∀ ε > 0, {t ∈ K : |f (t)| ≥ ε} compact}
– with the supremum norm. For f ∈ C0(K ), Mf := {t ∈ K : |f (t)| = ‖f ‖}(nonempty and compact).
Theorem
Let f , g ∈ C0(K ), f 6= 0 6= g . Assume that Mf is connected. Then, thefollowing conditions are equivalent:
(a) f⊥εBg ,
(b) ∃ t1 ∈ Mf : |g(t1)| ≤ ε‖g‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 12 / 28
Approximate orthogonality in C0(K )
L(H) Let K be a locally compact topological space.
C0(K ) := {f : K → R cont. : ∀ ε > 0, {t ∈ K : |f (t)| ≥ ε} compact}
– with the supremum norm. For f ∈ C0(K ), Mf := {t ∈ K : |f (t)| = ‖f ‖}(nonempty and compact).
Theorem
Let f , g ∈ C0(K ), f 6= 0 6= g . Assume that Mf is connected. Then, thefollowing conditions are equivalent:
(a) f⊥εBg ,
(b) ∃ t1 ∈ Mf : |g(t1)| ≤ ε‖g‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 12 / 28
Approximate orthogonality in C0(K )
L(H) Let K be a locally compact topological space.
C0(K ) := {f : K → R cont. : ∀ ε > 0, {t ∈ K : |f (t)| ≥ ε} compact}
– with the supremum norm. For f ∈ C0(K ), Mf := {t ∈ K : |f (t)| = ‖f ‖}(nonempty and compact).
Theorem
Let f , g ∈ C0(K ), f 6= 0 6= g . Assume that Mf is connected. Then, thefollowing conditions are equivalent:
(a) f⊥εBg ,
(b) ∃ t1 ∈ Mf : |g(t1)| ≤ ε‖g‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 12 / 28
Approximate orthogonality in C0(K )
L(H) Let K be a locally compact topological space.
C0(K ) := {f : K → R cont. : ∀ ε > 0, {t ∈ K : |f (t)| ≥ ε} compact}
– with the supremum norm. For f ∈ C0(K ), Mf := {t ∈ K : |f (t)| = ‖f ‖}(nonempty and compact).
Theorem
Let f , g ∈ C0(K ), f 6= 0 6= g . Assume that Mf is connected. Then, thefollowing conditions are equivalent:
(a) f⊥εBg ,
(b) ∃ t1 ∈ Mf : |g(t1)| ≤ ε‖g‖.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 12 / 28
Approximate symmetry of B-orthogonality
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 13 / 28
Symmetry of ⊥B
Birkhoff orthogonality ⊥B is generally not symmetric.
x
y
x+λy
y+λx
Figure: R2 with the maximum norm; x⊥By , y 6⊥Bx
If dimX ≥ 3 and ⊥B – symmetric, then X is an inner product space.If dimX = 2, then the symmetry of ⊥B is possible even if the norm doesnot come from an inner product (Radon plane).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 14 / 28
Symmetry of ⊥B
Birkhoff orthogonality ⊥B is generally not symmetric.
x
y
x+λy
y+λx
Figure: R2 with the maximum norm; x⊥By , y 6⊥Bx
If dimX ≥ 3 and ⊥B – symmetric, then X is an inner product space.If dimX = 2, then the symmetry of ⊥B is possible even if the norm doesnot come from an inner product (Radon plane).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 14 / 28
Symmetry of ⊥B
Birkhoff orthogonality ⊥B is generally not symmetric.
x
y
x+λy
y+λx
Figure: R2 with the maximum norm; x⊥By , y 6⊥Bx
If dimX ≥ 3 and ⊥B – symmetric, then X is an inner product space.If dimX = 2, then the symmetry of ⊥B is possible even if the norm doesnot come from an inner product (Radon plane).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 14 / 28
Symmetry of ⊥B
Birkhoff orthogonality ⊥B is generally not symmetric.
x
y
x+λy
y+λx
Figure: R2 with the maximum norm; x⊥By , y 6⊥Bx
If dimX ≥ 3 and ⊥B – symmetric, then X is an inner product space.
If dimX = 2, then the symmetry of ⊥B is possible even if the norm doesnot come from an inner product (Radon plane).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 14 / 28
Symmetry of ⊥B
Birkhoff orthogonality ⊥B is generally not symmetric.
x
y
x+λy
y+λx
Figure: R2 with the maximum norm; x⊥By , y 6⊥Bx
If dimX ≥ 3 and ⊥B – symmetric, then X is an inner product space.If dimX = 2, then the symmetry of ⊥B is possible even if the norm doesnot come from an inner product (Radon plane).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 14 / 28
Approximate symmetry of ⊥B
J. Chmielinski, P. Wojcik, Approximate symmetry of the Birkhofforthogonality, J. Math. Anal. Appl. 461 (2018), 625–640.
Definition
The Birkhoff orthogonality relation in a normed space X is calledε-symmetric (for some ε ∈ [0, 1)), if for any x , y ∈ X :
x⊥By =⇒ y⊥εBx .
⊥B is ε-symmetric for some ε ∈ [0, 1) if and only if:
x⊥By =⇒ ∃ z ∈ Lin {x , y} : y⊥Bz , ‖z − x‖ ≤ ε‖x‖.
The approximate symmetry of ⊥B does not imply that the norm comesfrom an inner product (even if dimX ≥ 3).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 15 / 28
Approximate symmetry of ⊥B
J. Chmielinski, P. Wojcik, Approximate symmetry of the Birkhofforthogonality, J. Math. Anal. Appl. 461 (2018), 625–640.
Definition
The Birkhoff orthogonality relation in a normed space X is calledε-symmetric (for some ε ∈ [0, 1)), if for any x , y ∈ X :
x⊥By =⇒ y⊥εBx .
⊥B is ε-symmetric for some ε ∈ [0, 1) if and only if:
x⊥By =⇒ ∃ z ∈ Lin {x , y} : y⊥Bz , ‖z − x‖ ≤ ε‖x‖.
The approximate symmetry of ⊥B does not imply that the norm comesfrom an inner product (even if dimX ≥ 3).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 15 / 28
Approximate symmetry of ⊥B
J. Chmielinski, P. Wojcik, Approximate symmetry of the Birkhofforthogonality, J. Math. Anal. Appl. 461 (2018), 625–640.
Definition
The Birkhoff orthogonality relation in a normed space X is calledε-symmetric (for some ε ∈ [0, 1)), if for any x , y ∈ X :
x⊥By =⇒ y⊥εBx .
⊥B is ε-symmetric for some ε ∈ [0, 1) if and only if:
x⊥By =⇒ ∃ z ∈ Lin {x , y} : y⊥Bz , ‖z − x‖ ≤ ε‖x‖.
The approximate symmetry of ⊥B does not imply that the norm comesfrom an inner product (even if dimX ≥ 3).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 15 / 28
Approximate symmetry of ⊥B
J. Chmielinski, P. Wojcik, Approximate symmetry of the Birkhofforthogonality, J. Math. Anal. Appl. 461 (2018), 625–640.
Definition
The Birkhoff orthogonality relation in a normed space X is calledε-symmetric (for some ε ∈ [0, 1)), if for any x , y ∈ X :
x⊥By =⇒ y⊥εBx .
⊥B is ε-symmetric for some ε ∈ [0, 1) if and only if:
x⊥By =⇒ ∃ z ∈ Lin {x , y} : y⊥Bz , ‖z − x‖ ≤ ε‖x‖.
The approximate symmetry of ⊥B does not imply that the norm comesfrom an inner product (even if dimX ≥ 3).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 15 / 28
Approximate symmetry of ⊥B
J. Chmielinski, P. Wojcik, Approximate symmetry of the Birkhofforthogonality, J. Math. Anal. Appl. 461 (2018), 625–640.
Definition
The Birkhoff orthogonality relation in a normed space X is calledε-symmetric (for some ε ∈ [0, 1)), if for any x , y ∈ X :
x⊥By =⇒ y⊥εBx .
⊥B is ε-symmetric for some ε ∈ [0, 1) if and only if:
x⊥By =⇒ ∃ z ∈ Lin {x , y} : y⊥Bz , ‖z − x‖ ≤ ε‖x‖.
The approximate symmetry of ⊥B does not imply that the norm comesfrom an inner product (even if dimX ≥ 3).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 15 / 28
Sufficient conditions for approximate symmetry of ⊥B
A modulus of convexity of a normed space X , δX : [0, 2]→ [0, 1]:
δX (ε) := inf{
1−∥∥∥x + y
2
∥∥∥ : ‖x‖ ≤ 1, ‖y‖ ≤ 1, ‖x − y‖ ≥ ε}.
Theorem
If δX (1) > 0 and 1− 2δX (1) ≤ ε < 1, relation ⊥B is ε-symmetric.
Corollary
Suppose that for any ε ∈ [0, 1) the relation ⊥B is not ε-symmetric. Then
ε0(X ) := sup{ε ∈ [0, 2] : δX (ε) = 0} ≥ 1.
Moreover, if X is finite-dimensional, then R(X ) ≥ 1.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 16 / 28
Sufficient conditions for approximate symmetry of ⊥B
A modulus of convexity of a normed space X , δX : [0, 2]→ [0, 1]:
δX (ε) := inf{
1−∥∥∥x + y
2
∥∥∥ : ‖x‖ ≤ 1, ‖y‖ ≤ 1, ‖x − y‖ ≥ ε}.
Theorem
If δX (1) > 0 and 1− 2δX (1) ≤ ε < 1, relation ⊥B is ε-symmetric.
Corollary
Suppose that for any ε ∈ [0, 1) the relation ⊥B is not ε-symmetric. Then
ε0(X ) := sup{ε ∈ [0, 2] : δX (ε) = 0} ≥ 1.
Moreover, if X is finite-dimensional, then R(X ) ≥ 1.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 16 / 28
Sufficient conditions for approximate symmetry of ⊥B
A modulus of convexity of a normed space X , δX : [0, 2]→ [0, 1]:
δX (ε) := inf{
1−∥∥∥x + y
2
∥∥∥ : ‖x‖ ≤ 1, ‖y‖ ≤ 1, ‖x − y‖ ≥ ε}.
Theorem
If δX (1) > 0 and 1− 2δX (1) ≤ ε < 1, relation ⊥B is ε-symmetric.
Corollary
Suppose that for any ε ∈ [0, 1) the relation ⊥B is not ε-symmetric. Then
ε0(X ) := sup{ε ∈ [0, 2] : δX (ε) = 0} ≥ 1.
Moreover, if X is finite-dimensional, then R(X ) ≥ 1.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 16 / 28
Sufficient conditions for approximate symmetry of ⊥B
A modulus of convexity of a normed space X , δX : [0, 2]→ [0, 1]:
δX (ε) := inf{
1−∥∥∥x + y
2
∥∥∥ : ‖x‖ ≤ 1, ‖y‖ ≤ 1, ‖x − y‖ ≥ ε}.
Theorem
If δX (1) > 0 and 1− 2δX (1) ≤ ε < 1, relation ⊥B is ε-symmetric.
Corollary
Suppose that for any ε ∈ [0, 1) the relation ⊥B is not ε-symmetric. Then
ε0(X ) := sup{ε ∈ [0, 2] : δX (ε) = 0} ≥ 1.
Moreover, if X is finite-dimensional, then R(X ) ≥ 1.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 16 / 28
Sufficient conditions for approximate symmetry of ⊥B
Theorem
Let X be a real, uniformly convex normed space.Then, ⊥B is approximately-symmetric.
Theorem
Let X be a finite-dimensional real smooth normed space.Then, ⊥B is approximately-symmetric.
Theorem
Let X be a real uniformly convex and smooth Banach space. Then, theBirkhoff orthogonality relations in X , X ∗ and X ∗∗ are approximatelysymmetric (actually, with the same ε).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 17 / 28
Sufficient conditions for approximate symmetry of ⊥B
Theorem
Let X be a real, uniformly convex normed space.Then, ⊥B is approximately-symmetric.
Theorem
Let X be a finite-dimensional real smooth normed space.Then, ⊥B is approximately-symmetric.
Theorem
Let X be a real uniformly convex and smooth Banach space. Then, theBirkhoff orthogonality relations in X , X ∗ and X ∗∗ are approximatelysymmetric (actually, with the same ε).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 17 / 28
Sufficient conditions for approximate symmetry of ⊥B
Theorem
Let X be a real, uniformly convex normed space.Then, ⊥B is approximately-symmetric.
Theorem
Let X be a finite-dimensional real smooth normed space.Then, ⊥B is approximately-symmetric.
Theorem
Let X be a real uniformly convex and smooth Banach space. Then, theBirkhoff orthogonality relations in X , X ∗ and X ∗∗ are approximatelysymmetric (actually, with the same ε).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 17 / 28
There are spaces in which the Birkhoff orthogonality is not approximatelysymmetric, i.e., for any ε ∈ [0, 1), ⊥B is not ε-symmetric.
Example
X = R2 with the maximum norm.
x
yz
y+λx
y + λz
Figure: x⊥By , y 6⊥Bx , y 6⊥Bz , y 6⊥εBx
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 18 / 28
There are spaces in which the Birkhoff orthogonality is not approximatelysymmetric, i.e., for any ε ∈ [0, 1), ⊥B is not ε-symmetric.
Example
X = R2 with the maximum norm.
x
yz
y+λx
y + λz
Figure: x⊥By , y 6⊥Bx , y 6⊥Bz , y 6⊥εBx
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 18 / 28
There are spaces in which the Birkhoff orthogonality is not approximatelysymmetric, i.e., for any ε ∈ [0, 1), ⊥B is not ε-symmetric.
Example
X = R2 with the maximum norm.
x
yz
y+λx
y + λz
Figure: x⊥By , y 6⊥Bx , y 6⊥Bz , y 6⊥εBx
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 18 / 28
Geometrical properties connected with approximatesymmetry of B-orthogonality
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 19 / 28
R(X ) := sup{‖x − y‖ : conv {x , y} ⊂ SX}.
We consider the following property of X :
x , y ∈ X , x 6= y , conv {x , y} ⊂ SX =⇒ X is smooth at x − y . (∗)
Examples
Each smooth or strictly convex space satisfies (∗).R2 with the supremum norm (which is neither strictly convex nor smooth)also satisfies (∗).X = R2 with the norm for which the unit ball is a symmetric polygon suchthat sides are not parallel to diagonals, the condition (∗) is satisfied.
Theorem
Let X be a two-dimensional strictly convex normed space and let Y be astrictly convex and smooth normed space. Then the space L (X ,Y )satisfies (∗).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 20 / 28
R(X ) := sup{‖x − y‖ : conv {x , y} ⊂ SX}.
We consider the following property of X :
x , y ∈ X , x 6= y , conv {x , y} ⊂ SX =⇒ X is smooth at x − y . (∗)
Examples
Each smooth or strictly convex space satisfies (∗).R2 with the supremum norm (which is neither strictly convex nor smooth)also satisfies (∗).X = R2 with the norm for which the unit ball is a symmetric polygon suchthat sides are not parallel to diagonals, the condition (∗) is satisfied.
Theorem
Let X be a two-dimensional strictly convex normed space and let Y be astrictly convex and smooth normed space. Then the space L (X ,Y )satisfies (∗).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 20 / 28
R(X ) := sup{‖x − y‖ : conv {x , y} ⊂ SX}.
We consider the following property of X :
x , y ∈ X , x 6= y , conv {x , y} ⊂ SX =⇒ X is smooth at x − y . (∗)
Examples
Each smooth or strictly convex space satisfies (∗).
R2 with the supremum norm (which is neither strictly convex nor smooth)also satisfies (∗).X = R2 with the norm for which the unit ball is a symmetric polygon suchthat sides are not parallel to diagonals, the condition (∗) is satisfied.
Theorem
Let X be a two-dimensional strictly convex normed space and let Y be astrictly convex and smooth normed space. Then the space L (X ,Y )satisfies (∗).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 20 / 28
R(X ) := sup{‖x − y‖ : conv {x , y} ⊂ SX}.
We consider the following property of X :
x , y ∈ X , x 6= y , conv {x , y} ⊂ SX =⇒ X is smooth at x − y . (∗)
Examples
Each smooth or strictly convex space satisfies (∗).R2 with the supremum norm (which is neither strictly convex nor smooth)also satisfies (∗).
X = R2 with the norm for which the unit ball is a symmetric polygon suchthat sides are not parallel to diagonals, the condition (∗) is satisfied.
Theorem
Let X be a two-dimensional strictly convex normed space and let Y be astrictly convex and smooth normed space. Then the space L (X ,Y )satisfies (∗).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 20 / 28
R(X ) := sup{‖x − y‖ : conv {x , y} ⊂ SX}.
We consider the following property of X :
x , y ∈ X , x 6= y , conv {x , y} ⊂ SX =⇒ X is smooth at x − y . (∗)
Examples
Each smooth or strictly convex space satisfies (∗).R2 with the supremum norm (which is neither strictly convex nor smooth)also satisfies (∗).X = R2 with the norm for which the unit ball is a symmetric polygon suchthat sides are not parallel to diagonals, the condition (∗) is satisfied.
Theorem
Let X be a two-dimensional strictly convex normed space and let Y be astrictly convex and smooth normed space. Then the space L (X ,Y )satisfies (∗).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 20 / 28
R(X ) := sup{‖x − y‖ : conv {x , y} ⊂ SX}.
We consider the following property of X :
x , y ∈ X , x 6= y , conv {x , y} ⊂ SX =⇒ X is smooth at x − y . (∗)
Examples
Each smooth or strictly convex space satisfies (∗).R2 with the supremum norm (which is neither strictly convex nor smooth)also satisfies (∗).X = R2 with the norm for which the unit ball is a symmetric polygon suchthat sides are not parallel to diagonals, the condition (∗) is satisfied.
Theorem
Let X be a two-dimensional strictly convex normed space and let Y be astrictly convex and smooth normed space. Then the space L (X ,Y )satisfies (∗).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 20 / 28
Theorem
Let X be a real normed space satisfying (∗) and let ε ∈ (0, 1). If theorthogonality relation ⊥B in X is ε-symmetric, then R(X ) ≤ 2ε.
Corollary
If X is a real normed space satisfying (∗) and R(X ) = 2, then the Birkhofforthogonality in X is not approximately symmetric.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 21 / 28
Theorem
Let X be a real normed space satisfying (∗) and let ε ∈ (0, 1). If theorthogonality relation ⊥B in X is ε-symmetric, then R(X ) ≤ 2ε.
Corollary
If X is a real normed space satisfying (∗) and R(X ) = 2, then the Birkhofforthogonality in X is not approximately symmetric.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 21 / 28
Symmetry constant S(X )
Definition
S(X ) := inf{ε ∈ [0, 1] : ∀ x , y ∈ X x⊥By ⇒ y⊥εBx}.
S(X ) ∈ [0, 1]S(X ) = 0 means that ⊥B is symmetric.S(X ) = 1 means that ⊥B is not approximately symmetric.
S(X ) = sup{S(X0) : X0 is a two-dimensional subspace of X}.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 22 / 28
Symmetry constant S(X )
Definition
S(X ) := inf{ε ∈ [0, 1] : ∀ x , y ∈ X x⊥By ⇒ y⊥εBx}.
S(X ) ∈ [0, 1]S(X ) = 0 means that ⊥B is symmetric.S(X ) = 1 means that ⊥B is not approximately symmetric.
S(X ) = sup{S(X0) : X0 is a two-dimensional subspace of X}.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 22 / 28
Symmetry constant S(X )
Definition
S(X ) := inf{ε ∈ [0, 1] : ∀ x , y ∈ X x⊥By ⇒ y⊥εBx}.
S(X ) ∈ [0, 1]
S(X ) = 0 means that ⊥B is symmetric.S(X ) = 1 means that ⊥B is not approximately symmetric.
S(X ) = sup{S(X0) : X0 is a two-dimensional subspace of X}.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 22 / 28
Symmetry constant S(X )
Definition
S(X ) := inf{ε ∈ [0, 1] : ∀ x , y ∈ X x⊥By ⇒ y⊥εBx}.
S(X ) ∈ [0, 1]S(X ) = 0 means that ⊥B is symmetric.
S(X ) = 1 means that ⊥B is not approximately symmetric.
S(X ) = sup{S(X0) : X0 is a two-dimensional subspace of X}.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 22 / 28
Symmetry constant S(X )
Definition
S(X ) := inf{ε ∈ [0, 1] : ∀ x , y ∈ X x⊥By ⇒ y⊥εBx}.
S(X ) ∈ [0, 1]S(X ) = 0 means that ⊥B is symmetric.S(X ) = 1 means that ⊥B is not approximately symmetric.
S(X ) = sup{S(X0) : X0 is a two-dimensional subspace of X}.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 22 / 28
Symmetry constant S(X )
Definition
S(X ) := inf{ε ∈ [0, 1] : ∀ x , y ∈ X x⊥By ⇒ y⊥εBx}.
S(X ) ∈ [0, 1]S(X ) = 0 means that ⊥B is symmetric.S(X ) = 1 means that ⊥B is not approximately symmetric.
S(X ) = sup{S(X0) : X0 is a two-dimensional subspace of X}.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 22 / 28
If X is uniformly convex, then S(X ) < 1 (the reverse is not true).
If X satisfies (∗), then1
2R(X ) ≤ S(X ).
If X is a real uniformly convex and smooth Banach space, then
S(X ) = S(X ∗) = S(X ∗∗) < 1.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 23 / 28
If X is uniformly convex, then S(X ) < 1 (the reverse is not true).
If X satisfies (∗), then1
2R(X ) ≤ S(X ).
If X is a real uniformly convex and smooth Banach space, then
S(X ) = S(X ∗) = S(X ∗∗) < 1.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 23 / 28
If X is uniformly convex, then S(X ) < 1 (the reverse is not true).
If X satisfies (∗), then1
2R(X ) ≤ S(X ).
If X is a real uniformly convex and smooth Banach space, then
S(X ) = S(X ∗) = S(X ∗∗) < 1.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 23 / 28
Example
Let X = R2 with an l1-l∞ norm.
For δ > 0 let Y = Yδ = R2 with the norm such that the unit sphere is ahexagon with vertices at
y1 = (1, 0), y2 = (0, 1), y3 = (−1− δ, 1− δ), y4 = (−1, 0),
y5 = (0,−1), y6 = (1 + δ,−1 + δ).
x1 = y1
x2 = y2
x3
y3
x4 = y4
x5 = y5 x6
y6
Figure: Unit spheres in X and Y
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 24 / 28
Example
Let X = R2 with an l1-l∞ norm.For δ > 0 let Y = Yδ = R2 with the norm such that the unit sphere is ahexagon with vertices at
y1 = (1, 0), y2 = (0, 1), y3 = (−1− δ, 1− δ), y4 = (−1, 0),
y5 = (0,−1), y6 = (1 + δ,−1 + δ).
x1 = y1
x2 = y2
x3
y3
x4 = y4
x5 = y5 x6
y6
Figure: Unit spheres in X and Y
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 24 / 28
Example
Let X = R2 with an l1-l∞ norm.For δ > 0 let Y = Yδ = R2 with the norm such that the unit sphere is ahexagon with vertices at
y1 = (1, 0), y2 = (0, 1), y3 = (−1− δ, 1− δ), y4 = (−1, 0),
y5 = (0,−1), y6 = (1 + δ,−1 + δ).
x1 = y1
x2 = y2
x3
y3
x4 = y4
x5 = y5 x6
y6
Figure: Unit spheres in X and Y
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 24 / 28
It can be checked that the Banach-Mazur distance d(X ,Y ) can bearbitrarily close to 1; namely:
d(X ,Y ) ≤ 1 + δ
1− δ.
The space X is a Radon plane, therefore S(X ) = 0.
No matter how small is δ > 0, the space Y satisfies (∗), whence
S(Y ) ≥ 1
2R(Y ) >
1
2.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 25 / 28
It can be checked that the Banach-Mazur distance d(X ,Y ) can bearbitrarily close to 1; namely:
d(X ,Y ) ≤ 1 + δ
1− δ.
The space X is a Radon plane, therefore S(X ) = 0.
No matter how small is δ > 0, the space Y satisfies (∗), whence
S(Y ) ≥ 1
2R(Y ) >
1
2.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 25 / 28
It can be checked that the Banach-Mazur distance d(X ,Y ) can bearbitrarily close to 1; namely:
d(X ,Y ) ≤ 1 + δ
1− δ.
The space X is a Radon plane, therefore S(X ) = 0.
No matter how small is δ > 0, the space Y satisfies (∗)
, whence
S(Y ) ≥ 1
2R(Y ) >
1
2.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 25 / 28
It can be checked that the Banach-Mazur distance d(X ,Y ) can bearbitrarily close to 1; namely:
d(X ,Y ) ≤ 1 + δ
1− δ.
The space X is a Radon plane, therefore S(X ) = 0.
No matter how small is δ > 0, the space Y satisfies (∗), whence
S(Y ) ≥ 1
2R(Y ) >
1
2.
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 25 / 28
S(lpn ) (p > 1)
The Banach-Mazur distance between lpn and l2n is equal to:
d := d(lpn , l2n ) = n
∣∣∣ 1p− 1
2
∣∣∣.
We were able to estimate that for p > 1, sufficiently close to 2, we have
S(lpn ) ≤ max
{(2p −
(1 +
1
d2
)p) 1p
,
(2q −
(1 +
1
d2
)q) 1q
}
(with q such that 1p + 1
q = 1).If p → 2, then q → 2 and d → 1.Therefore
limp→2
S(lpn ) = 0 = S(l2n ).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 26 / 28
S(lpn ) (p > 1)
The Banach-Mazur distance between lpn and l2n is equal to:
d := d(lpn , l2n ) = n
∣∣∣ 1p− 1
2
∣∣∣.
We were able to estimate that for p > 1, sufficiently close to 2, we have
S(lpn ) ≤ max
{(2p −
(1 +
1
d2
)p) 1p
,
(2q −
(1 +
1
d2
)q) 1q
}
(with q such that 1p + 1
q = 1).If p → 2, then q → 2 and d → 1.Therefore
limp→2
S(lpn ) = 0 = S(l2n ).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 26 / 28
S(lpn ) (p > 1)
The Banach-Mazur distance between lpn and l2n is equal to:
d := d(lpn , l2n ) = n
∣∣∣ 1p− 1
2
∣∣∣.
We were able to estimate that for p > 1, sufficiently close to 2, we have
S(lpn ) ≤ max
{(2p −
(1 +
1
d2
)p) 1p
,
(2q −
(1 +
1
d2
)q) 1q
}
(with q such that 1p + 1
q = 1).If p → 2, then q → 2 and d → 1.Therefore
limp→2
S(lpn ) = 0 = S(l2n ).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 26 / 28
S(lpn ) (p > 1)
The Banach-Mazur distance between lpn and l2n is equal to:
d := d(lpn , l2n ) = n
∣∣∣ 1p− 1
2
∣∣∣.
We were able to estimate that for p > 1, sufficiently close to 2, we have
S(lpn ) ≤ max
{(2p −
(1 +
1
d2
)p) 1p
,
(2q −
(1 +
1
d2
)q) 1q
}
(with q such that 1p + 1
q = 1).
If p → 2, then q → 2 and d → 1.Therefore
limp→2
S(lpn ) = 0 = S(l2n ).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 26 / 28
S(lpn ) (p > 1)
The Banach-Mazur distance between lpn and l2n is equal to:
d := d(lpn , l2n ) = n
∣∣∣ 1p− 1
2
∣∣∣.
We were able to estimate that for p > 1, sufficiently close to 2, we have
S(lpn ) ≤ max
{(2p −
(1 +
1
d2
)p) 1p
,
(2q −
(1 +
1
d2
)q) 1q
}
(with q such that 1p + 1
q = 1).If p → 2, then q → 2 and d → 1.
Thereforelimp→2
S(lpn ) = 0 = S(l2n ).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 26 / 28
S(lpn ) (p > 1)
The Banach-Mazur distance between lpn and l2n is equal to:
d := d(lpn , l2n ) = n
∣∣∣ 1p− 1
2
∣∣∣.
We were able to estimate that for p > 1, sufficiently close to 2, we have
S(lpn ) ≤ max
{(2p −
(1 +
1
d2
)p) 1p
,
(2q −
(1 +
1
d2
)q) 1q
}
(with q such that 1p + 1
q = 1).If p → 2, then q → 2 and d → 1.Therefore
limp→2
S(lpn ) = 0 = S(l2n ).
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 26 / 28
Thank you!
J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 27 / 28
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J. Chmielinski (Krakow, Poland) Approximate Birkhoff orthogonality Lviv 2019 28 / 28