Classification of Nonselfadjoint Operators

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Operator System Equivalence Irreducible Periodic Weighted Shift Operators Subspaces Determined by Single-Generated Algebras Concluding Remarks

Classiﬁcation of Nonselfadjoint Operators via Operator System Equivalence

Douglas Farenick, with M. Argerami, M. Mastnak, A. Popov

Workshop on Numerical Ranges and Applications Max-Planck Institute for Quantum Optics Technical University of Munich 14 June 2018

Douglas Farenick, with M. Argerami, M. Mastnak, A. Popov Classiﬁcation of Nonselfadjoint Operators Operator System Equivalence Irreducible Periodic Weighted Shift Operators Subspaces Determined by Single-Generated Algebras Concluding Remarks

Operator System Equivalence

Deﬁnition (Toeplitz-Hausdorﬀ, 1918) The (spatial) numerical range of a bounded linear operator T on a complex Hilbert space H is the set

Ws(T ) = {hT ξ, ξi : ξ ∈ H, kξk = 1}.

Deﬁnition (Banach Algebra Community, 1950s) The numerical range of an element x in a unital Banach algebra A is the set

∗ W (x) = {ϕ(x): ϕ ∈ A , kϕk = ϕ(1A) = 1}.

I By the Hahn-Banach Extension Theorem, if X is a subspace of a unital Banach algebra A such that 1A ∈ X, then

∗ W (x) = {ϕ(x): ϕ ∈ X , kϕk = ϕ(1A) = 1},

for every x ∈ X.

I Thus, W (x) is determined by certain elements of the unit sphere in the dual space X∗ of any subspace of A that contains 1A and x, such as the 2-dimensional subspace Span {1A, x}. Hence,

W (x) = {λ ∈ C: |αλ + β| ≤ kαx + β1Ak, ∀ α, β ∈ C}.

I In the case of B(H), the spatial numerical range and numerical range of T ∈ B(H) are related via:

W (T ) = Ws(T ),

If H has ﬁnite dimension, then W (T ) = Ws(T ) because Ws(T ) is already a closed set. ∗ I Because B(H) has an involution T 7→ T , it is more natural to consider W (T ) in terms of the subspace

∗ ST = Span {1, T , T }, where 1 denotes the identity operator ξ 7→ ξ on H. That is,

∗ W (T ) = {ϕ(T ): ϕ ∈ (ST ) , kϕk = ϕ(1) = 1}.

∗ I A subspace of the form ST = Span {1, T , T }, for some T ∈ B(H), is spanned by its positive elements

I The conditions kϕk = ϕ(1) = 1 are equivalent to ϕ(1) = 1 and ϕ(A) ≥ 0 for every positive A ∈ ST .

I If by a state on ST we mean a linear functional ϕ such that ϕ(1) = 1 and ϕ(A) ≥ 0 for every positive A ∈ ST , then

W (T ) = {ϕ(T ): ϕ is a state on ST }.

I In place of Hahn-Banach Extension Theorem, we have the Krein Extension Theorem: every state on ST extends to a state on B(H). Thus, the deﬁnition of W (T ) above does not depend on the choice of unital ∗-closed subspace that contains 1, T , and T ∗.

Douglas Farenick, with M. Argerami, M. Mastnak, A. Popov Classiﬁcation of Nonselfadjoint Operators Now let us replace the scalars λ, α, β, γ by n × n complex matrices Λ, A, B, and C

Operator System Equivalence Irreducible Periodic Weighted Shift Operators Subspaces Determined by Single-Generated Algebras Concluding Remarks Remarks: numerical range for Hilbert space operators

∗ In a subspace of the form ST = Span {1, T , T }, elements X are given by X = α1 + βT + γT ∗. Thus, λ ∈ W (T ) if and only if

|α + βλ + γλ| ≤ kα1 + βT + γT ∗k,

for a scalars α, β, γ.

∗ In a subspace of the form ST = Span {1, T , T }, elements X are given by X = α1 + βT + γT ∗. Thus, λ ∈ W (T ) if and only if

|α + βλ + γλ| ≤ kα1 + βT + γT ∗k,

for a scalars α, β, γ. Now let us replace the scalars λ, α, β, γ by n × n complex matrices Λ, A, B, and C

Deﬁnition The n-th matricial range of T ∈ B(H) is the subset Wn(T ) of all n × n complex matrices Λ such that

∗ ∗ kA ⊗ 1n + B ⊗ Λ + C ⊗ Λ k ≤ kA ⊗ 1 + B ⊗ T + C ⊗ T k,

for all n × n complex matrices A, B, and C. (Here, 1n is the n × n identity matrix and 1 is the identity operator on H.) Notes:

I 1n is the n × n identity matrix and 1 is the identity operator on H

I the norm on each of Mn(C) ⊗ Mn(C) and Mn(C) ⊗ B(H) is the “operator norm”

Douglas Farenick, with M. Argerami, M. Mastnak, A. Popov Classiﬁcation of Nonselfadjoint Operators ∗ I Suﬃcient condition for S 'os T : S = U TU, for some unitary U

I Necessary condition for S 'os T : Wn(S) = Wn(T ), for every n

Operator System Equivalence Irreducible Periodic Weighted Shift Operators Subspaces Determined by Single-Generated Algebras Concluding Remarks A Notion of Equivalence for Operators

Deﬁnition If S, T ∈ B(H), then S and T are operator system equivalent, denoted by S 'os T , if

∗ ∗ kA ⊗ 1n + B ⊗ S + C ⊗ S k = kA ⊗ 1 + B ⊗ T + C ⊗ T k,

for all n × n complex matrices A, B, and C, and all positive integers n.

Douglas Farenick, with M. Argerami, M. Mastnak, A. Popov Classiﬁcation of Nonselfadjoint Operators I Necessary condition for S 'os T : Wn(S) = Wn(T ), for every n

Operator System Equivalence Irreducible Periodic Weighted Shift Operators Subspaces Determined by Single-Generated Algebras Concluding Remarks A Notion of Equivalence for Operators

Deﬁnition If S, T ∈ B(H), then S and T are operator system equivalent, denoted by S 'os T , if

∗ ∗ kA ⊗ 1n + B ⊗ S + C ⊗ S k = kA ⊗ 1 + B ⊗ T + C ⊗ T k,

for all n × n complex matrices A, B, and C, and all positive integers n.

∗ I Suﬃcient condition for S 'os T : S = U TU, for some unitary U

Deﬁnition If S, T ∈ B(H), then S and T are operator system equivalent, denoted by S 'os T , if

∗ ∗ kA ⊗ 1n + B ⊗ S + C ⊗ S k = kA ⊗ 1 + B ⊗ T + C ⊗ T k,

for all n × n complex matrices A, B, and C, and all positive integers n.

∗ I Suﬃcient condition for S 'os T : S = U TU, for some unitary U

I Necessary condition for S 'os T : Wn(S) = Wn(T ), for every n

Douglas Farenick, with M. Argerami, M. Mastnak, A. Popov Classiﬁcation of Nonselfadjoint Operators 2. S and T have the same numerical ranges

Note: the numerical range does not determine a hermitian operator up to unitary (or approximate-unitary) equivalence, but it does determine a hermitian operator up to operator-system equivalence

Operator System Equivalence Irreducible Periodic Weighted Shift Operators Subspaces Determined by Single-Generated Algebras Concluding Remarks Examples of Operator System Equivalence

Example (1) The following statements are equivalent for operators S and T such that S∗ = S:

1. S 'os T

Example (1) The following statements are equivalent for operators S and T such that S∗ = S:

1. S 'os T 2. S and T have the same numerical ranges

Note: the numerical range does not determine a hermitian operator up to unitary (or approximate-unitary) equivalence, but it does determine a hermitian operator up to operator-system equivalence

Example (2) If B is the forward shift (a.k.a bilateral shift) operator on `2(Z), and if S is the forward shift (a.k.a unilateral shift) operator on `2(N), then B 'os S.

Note: B is unitary and S is not unitary; thus, B and S cannot be unitarily equivalent. Therefore, operator system equivalence is a weaker notion than unitary equivalence.

Douglas Farenick, with M. Argerami, M. Mastnak, A. Popov Classiﬁcation of Nonselfadjoint Operators Theorem (Arveson, 1972) The following statements are equivalent for irreducible compact operators: 1. S and T are operator system equivalent 2. S and T are unitarily equivalent

Operator System Equivalence Irreducible Periodic Weighted Shift Operators Subspaces Determined by Single-Generated Algebras Concluding Remarks Arveson’s Theorem

Deﬁnition An operator T is irreducible if T is not unitarily equivalent to an operator of the form T1 ⊕ T2.

Douglas Farenick, with M. Argerami, M. Mastnak, A. Popov Classiﬁcation of Nonselfadjoint Operators 2. S and T are unitarily equivalent

Operator System Equivalence Irreducible Periodic Weighted Shift Operators Subspaces Determined by Single-Generated Algebras Concluding Remarks Arveson’s Theorem

Deﬁnition An operator T is irreducible if T is not unitarily equivalent to an operator of the form T1 ⊕ T2.

Theorem (Arveson, 1972) The following statements are equivalent for irreducible compact operators: 1. S and T are operator system equivalent

Deﬁnition An operator T is irreducible if T is not unitarily equivalent to an operator of the form T1 ⊕ T2.

Theorem (Arveson, 1972) The following statements are equivalent for irreducible compact operators: 1. S and T are operator system equivalent 2. S and T are unitarily equivalent

Theorem (Hamana, 1979) If T ∈ B(H), then there exists a unital C∗-algebra, denoted by ∗ ∗ Ce(T ), and a unital complete isometry ιe :ST → Ce(T ) such that ∗ 1. Ce(T ) is generated by the range of ιe, and 2. if φ :ST → A is a unital complete isometry such that A is generated by the range of φ, then there exists a surjective ∗ ∗-homomorphism π :A → Ce(T ) such that π ◦ φ = ιe. Deﬁnition If S and T are ∗-closed subspaces of B(H) in which 1 ∈ S and 1 ∈ T, then a linear map φ :S → T is a unital complete isometry if

[φ(S )]n = [S ]n , ∀ S ∈ S, ∀ n ∈ N ij i,j=1 ij i,j=1 i,j

Example (1) ∗ ∗ If T = T , then Ce(T ) = C ⊕ C Example (2) For each λ ∈ C, let

 0 1 0  Tλ =  0 0 0  . 0 0 λ

Then: ∗ M2(C) if |λ| ≤ 1/2 Ce(Tλ) = . M2(C) ⊕ C if |λ| > 1/2

Notation: Jk (λ) denotes a k × k Jordan block with eigenvalue λ

Theorem (Argerami, Farenick) n M If J = (Jmk (λk ) ⊗ 1dk ), where each pair (mk , λk ) unique, and k=1 if each λk is a real number, then the following statements are equivalent: ∗ 1. Ce(J) is abelian 2. m1 = ··· = mk = 1

Douglas Farenick, with M. Argerami, M. Mastnak, A. Popov Classiﬁcation of Nonselfadjoint Operators ∗ 2. S 'Ce T

Thus, operator system equivalence amounts to the identiﬁcation of ∗ Ce(T ) up to ∗-isomorphism.

Operator System Equivalence Irreducible Periodic Weighted Shift Operators Subspaces Determined by Single-Generated Algebras Concluding Remarks C∗-Envelope Equivalence

Deﬁnition ∗ If S, T ∈ B(H), then S and T are Ce-equivalent, denoted by ∗ S 'Ce T , if there exists a unital ∗-isomorphism ∗ ∗ % : Ce(S) → Ce(T ) such that %(S) = T . Theorem The following statements are equivalent for operators S and T :

1. S 'os T

Douglas Farenick, with M. Argerami, M. Mastnak, A. Popov Classiﬁcation of Nonselfadjoint Operators Thus, operator system equivalence amounts to the identiﬁcation of ∗ Ce(T ) up to ∗-isomorphism.

Operator System Equivalence Irreducible Periodic Weighted Shift Operators Subspaces Determined by Single-Generated Algebras Concluding Remarks C∗-Envelope Equivalence

1. S 'os T

∗ 2. S 'Ce T

1. S 'os T

∗ 2. S 'Ce T

Thus, operator system equivalence amounts to the identiﬁcation of ∗ Ce(T ) up to ∗-isomorphism.

Irreducible Periodic Weighted Shift Operators

Deﬁnition If ξ = (ξ1, . . . , ξd ) with ξj ∈ C \{0} for all j, then the irreducible weighted unilateral shift with weights ξ1, . . . , ξd is the operator W (ξ) on Cd+1 given by the matrix

 0 0   ξ 0   1   ..  W (ξ) =  ξ2 .  .    ..   . 0  ξd 0

Douglas Farenick, with M. Argerami, M. Mastnak, A. Popov Classiﬁcation of Nonselfadjoint Operators 2. W (ξ) = U∗W (η)U for some unitary U

3. |ξj | = |ηj | for all j

Operator System Equivalence Irreducible Periodic Weighted Shift Operators Subspaces Determined by Single-Generated Algebras Concluding Remarks Finite-Dimensional Hilbert Spaces

Theorem (Farenick, Gerasimova, Shvai, 2011) The C∗-envelope of an irreducible weighted unilateral shift acting d+1 d on C is Md+1(C). Furthermore, for any ξ, η ∈ C with nonzero entries, the following statements are equivalent:

1. W (ξ) 'os W (η)

Douglas Farenick, with M. Argerami, M. Mastnak, A. Popov Classiﬁcation of Nonselfadjoint Operators 3. |ξj | = |ηj | for all j

Operator System Equivalence Irreducible Periodic Weighted Shift Operators Subspaces Determined by Single-Generated Algebras Concluding Remarks Finite-Dimensional Hilbert Spaces

1. W (ξ) 'os W (η) 2. W (ξ) = U∗W (η)U for some unitary U

3. |ξj | = |ηj | for all j

Deﬁnition A weighted unilateral shift operator of period p is an operator W (ω) on `2(N) deﬁned on the standard orthonormal basis 2 {en : n ∈ N} of ` (N) by

W (ω)en = ωnen+1 , n ∈ N,

where the weight sequence ω = {ωn}n∈N for W consists of complex numbers satisfying

I supn |ωn| < ∞, and I ωn+p = ωn for every n ∈ N, where p is the least positive integer with this property

Remark: If ωn 6= 0 for all n, the W (ω) is an irreducible operator. Douglas Farenick, with M. Argerami, M. Mastnak, A. Popov Classiﬁcation of Nonselfadjoint Operators Operator System Equivalence Irreducible Periodic Weighted Shift Operators Subspaces Determined by Single-Generated Algebras Concluding Remarks Recall: the case p = 1

An irreducible periodic weighted unilateral shift operator is simply a scalar multiple ω1 of the (unweighted) unilateral shift. In this case, S 'os U, for every unitary operator U with σ(U) = ∂D. Thus, it may be more fruitful to determine those operators T for ∗ ∼ ∗ which Ce(T ) = Ce(S) via a unital ∗-isomorphism % in which ∗ %(T ) = S — that is, to determine T such that T 'Ce S. In the case p = 1, this possible because of the Gelfand theory for commutative C∗-algebras (even though the C∗-algebra generated by S is noncommutative).

Deﬁnition If ω = {ω1, . . . , ωp} is a sequence of complex numbers, then G(ω): ∂D → Mp(C) denotes the matrix-valued function given by   0 ωpz  ω 0   1   ..  G(ω)[z] =  ω2 .  , for z ∈ ∂D.    ..   . 0  ωp−1 0

Note: The algebra Mp (C(∂D)) of all continuous functions ∗ ∂D → Mp(C) is given by the C -algebra C(∂D) ⊗ Mp(C)

Douglas Farenick, with M. Argerami, M. Mastnak, A. Popov Classiﬁcation of Nonselfadjoint Operators ∗ 2. W (ω) 'Ce G(ω) Moreover, if one of the statements above holds, then

∗ ∼ ∗ ∼ ∼ Ce (W (ω)) = Ce (G(ω)) = C(∂D) ⊗ Mp(C) = Mp (C(∂D)) .

Operator System Equivalence Irreducible Periodic Weighted Shift Operators Subspaces Determined by Single-Generated Algebras Concluding Remarks Main Result

Theorem (Argerami, Farenick)

If ω = {ω1, . . . , ωp} is a sequence of nonzero complex numbers, and if there is at least one k with |ωk | 6∈ {|ωj | : j 6= k}, then the following statements are equivalent:

1. W (ω) 'os G(ω)

Douglas Farenick, with M. Argerami, M. Mastnak, A. Popov Classiﬁcation of Nonselfadjoint Operators Moreover, if one of the statements above holds, then

∗ ∼ ∗ ∼ ∼ Ce (W (ω)) = Ce (G(ω)) = C(∂D) ⊗ Mp(C) = Mp (C(∂D)) .

Operator System Equivalence Irreducible Periodic Weighted Shift Operators Subspaces Determined by Single-Generated Algebras Concluding Remarks Main Result

Theorem (Argerami, Farenick)

If ω = {ω1, . . . , ωp} is a sequence of nonzero complex numbers, and if there is at least one k with |ωk | 6∈ {|ωj | : j 6= k}, then the following statements are equivalent:

1. W (ω) 'os G(ω)

∗ 2. W (ω) 'Ce G(ω)

Theorem (Argerami, Farenick)

If ω = {ω1, . . . , ωp} is a sequence of nonzero complex numbers, and if there is at least one k with |ωk | 6∈ {|ωj | : j 6= k}, then the following statements are equivalent:

1. W (ω) 'os G(ω)

∗ 2. W (ω) 'Ce G(ω) Moreover, if one of the statements above holds, then

∗ ∼ ∗ ∼ ∼ Ce (W (ω)) = Ce (G(ω)) = C(∂D) ⊗ Mp(C) = Mp (C(∂D)) .

For a ﬁxed λ0 ∈ C of modulus |λ0| = 1, consider the matrix   0 ωpλ0  ω 0   1   ..  Ωλ = G(ω)[λ0] =  ω2 .  . 0    ..   . 0  ωp−1 0

I One can think of Ωλ0 as the result of evaluating “point evaluation” f 7→ f [λ0], for f ∈ Mp (C(∂D)), at f = G(ω)

I In particular, Ωλ0 ∈ Wp (G(ω)))

Douglas Farenick, with M. Argerami, M. Mastnak, A. Popov Classiﬁcation of Nonselfadjoint Operators I personal thought: the Tsai-Wu result is something that Hausdorﬀ and Toeplitz would have appreciated, 100 years ago

Operator System Equivalence Irreducible Periodic Weighted Shift Operators Subspaces Determined by Single-Generated Algebras Concluding Remarks Main Result: the use of numerical and matricial ranges

I Motivating idea from Gelfand theory: point evaluations on commutative C∗-algebras are extremal states

I Matrix case: there is a strong sense in which the “point

evaluation matricial state” Γz0 (f ) = f (z0) is extremal, leading ∗ to the fact that Ωλ0 is a C -extreme point of the matricial range Wp (G(ω))) of G(ω)

I crucial result of Tsai-Wu (2011): if r = w(Ωλ0 ) (numerical radius), then

k W (Ωλ0 ) ∩ {z : |z| = r} = {rζ λ0 : k = 1,..., p},

where ζ is a primitive p-th root of unity

I Motivating idea from Gelfand theory: point evaluations on commutative C∗-algebras are extremal states

I Matrix case: there is a strong sense in which the “point

evaluation matricial state” Γz0 (f ) = f (z0) is extremal, leading ∗ to the fact that Ωλ0 is a C -extreme point of the matricial range Wp (G(ω))) of G(ω)

I crucial result of Tsai-Wu (2011): if r = w(Ωλ0 ) (numerical radius), then

k W (Ωλ0 ) ∩ {z : |z| = r} = {rζ λ0 : k = 1,..., p},

where ζ is a primitive p-th root of unity I personal thought: the Tsai-Wu result is something that Hausdorﬀ and Toeplitz would have appreciated, 100 years ago

Subspaces Determined by Single-Generated Algebras

Let Alg(T ) = {f (T ): f is a complex polynomial}, which is a unital, abelian Banach algebra

Deﬁnition The algebra operator system determined by T is the unital ∗-closed subspace AT given by

∗ AT = {X + Y : X , Y ∈ Alg(T )}.

Although AT ⊇ ST , the algebra operator system determined by T carries more information about T than ST does.

If J = Jn(0), the n × n nilpotent Jordan block, then AJ is the unital ∗-closed subspace of all n × n Toeplitz1 matrices, which we denoted by Tn. Theorem (Farenick, Mastnak, Popov)

If ϕ : Tn → Mn(C) is a unital linear isometry, then there is a ∗ unitary matrix U such that ϕ(X ) = U XU for every X ∈ Tn. Note: ϕ maps n × n Toeplitz matrices to n × n matrices

1Yes, the same Toeplitz! Douglas Farenick, with M. Argerami, M. Mastnak, A. Popov Classiﬁcation of Nonselfadjoint Operators Operator System Equivalence Irreducible Periodic Weighted Shift Operators Subspaces Determined by Single-Generated Algebras Concluding Remarks Example: Finite Toeplitz Matrices

Theorem (Farenick, Mastnak, Popov)

If ϕ : Tn → Mm(C) is a unital completely isometric linear map, then n ≤ m and there are a unitary U ∈ Mm(C), a positive integer `, and a unital completely contractive map ψ : Tn → Mm−`n(C) such that ∗ ϕ(X ) = U ([X ⊗ I`] ⊕ ψ(X )) U,

for every X ∈ Tn. Remark: Allowing m 6= n forces us to move from isometries in the previous theorem to complete isometries in the theorem above

1. The numerical range lends itself to be deﬁned in terms of states on a low-dimensional unital subspace 2. Once deﬁned in terms of norms, one can consider “matrix-linear combinations” rather than classical linear combinations to arrive at the notion of a matricial range (there is a spatial approach, also, which I did not mention) 3. Operator system equivalence is a weaker notion than unitary equivalence, and has potential uses in describing various phenomena (e.g., the study of “clean POVMs” )

4. The notion of AT is motivated by Arveson’s seminal works in 1969/1972, where he studies nonselfadjoint operator algebras A by way of the analytic features of the subspace A + A∗

Thank you for attending WONRA 2018,

the 100th anniversary of the Toeplitz-Hausdorﬀ Theorem

Douglas Farenick, with M. Argerami, M. Mastnak, A. Popov Classiﬁcation of Nonselfadjoint Operators