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Classification of Nonselfadjoint Operators

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Classification of Nonselfadjoint Operators 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 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 Numerical Ranges 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 ) = fhT ξ; ξi : ξ 2 H; kξk = 1g: Definition (Banach Algebra Community, 1950s) The numerical range of an element x in a unital Banach algebra A is the set ∗ W (x) = f'(x): ' 2 A ; k'k = '(1A) = 1g: 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 Remarks I By the Hahn-Banach Extension Theorem, if X is a subspace of a unital Banach algebra A such that 1A 2 X, then ∗ W (x) = f'(x): ' 2 X ; k'k = '(1A) = 1g; for every x 2 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 f1A; xg. Hence, W (x) = fλ 2 C: jαλ + βj ≤ kαx + β1Ak; 8 α; β 2 Cg: 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 Remarks: numerical range for Hilbert space operators I In the case of B(H), the spatial numerical range and numerical range of T 2 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 f1; T ; T g; where 1 denotes the identity operator ξ 7! ξ on H. That is, ∗ W (T ) = f'(T ): ' 2 (ST ) ; k'k = '(1) = 1g: 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 Remarks: numerical range for Hilbert space operators ∗ I A subspace of the form ST = Span f1; T ; T g, for some T 2 B(H), is spanned by its positive elements I The conditions k'k = '(1) = 1 are equivalent to '(1) = 1 and '(A) ≥ 0 for every positive A 2 ST . I If by a state on ST we mean a linear functional ' such that '(1) = 1 and '(A) ≥ 0 for every positive A 2 ST , then W (T ) = f'(T ): ' is a state on ST g: 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 f1; T ; T g, elements X are given by X = α1 + βT + γT ∗: Thus, λ 2 W (T ) if and only if jα + βλ + γλj ≤ kα1 + βT + γT ∗k; for a scalars α, β, γ. 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 Remarks: numerical range for Hilbert space operators ∗ In a subspace of the form ST = Span f1; T ; T g, elements X are given by X = α1 + βT + γT ∗: Thus, λ 2 W (T ) if and only if jα + βλ + γλj ≤ 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 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 The matricial range for Hilbert space operators Definition The n-th matricial range of T 2 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 2 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 2 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 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 A Notion of Equivalence for Operators Definition If S; T 2 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 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 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 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 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 Examples of 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.
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