Extended tensor products and an operator-valued spectral mapping theorem

Kurbatov, V. G.; Kurbatova, I. V.

doi:10.1070/im2015v079n04abeh002759

Cited by 2 publications

(5 citation statements)

References 14 publications

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“…Note that the considerably more general operator functions on tensor products of spaces have been considered in [20] and references given therein. f (sI n +D) n .…”

Section: Applications To Operator Functionsmentioning

confidence: 99%

Similarity of Operators on Tensor Products of Spaces and Matrix Differential Operators

GIL’

2018

J. Aust. Math. Soc.

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Let ${\mathcal{H}}=\mathbb{C}^{n}\otimes {\mathcal{E}}$ be the tensor product of a Euclidean space $\mathbb{C}^{n}$ and a separable Hilbert space ${\mathcal{E}}$. Our main object is the operator $G=I_{n}\otimes S+A\otimes I_{{\mathcal{E}}}$, where $S$ is a normal operator in ${\mathcal{E}}$, $A$ is an $n\times n$ matrix, and $I_{n},I_{{\mathcal{E}}}$ are the unit operators in $\mathbb{C}^{n}$ and ${\mathcal{E}}$, respectively. Numerous differential operators with constant matrix coefficients are examples of operator $G$. In the present paper we show that $G$ is similar to an operator $M=I_{n}\otimes S+\hat{D}\times I_{{\mathcal{E}}}$ where $\hat{D}$ is a block matrix, each block of which has a unique eigenvalue. We also obtain a bound for the condition number. That bound enables us to establish norm estimates for functions of $G$, nonregular on the closed convex hull $\operatorname{co}(G)$ of the spectrum of $G$. The functions $G^{-\unicode[STIX]{x1D6FC}}\;(\unicode[STIX]{x1D6FC}>0)$ and $(\ln G)^{-1}$ are examples of such functions. In addition, in the appropriate situations we improve the previously published estimates for the resolvent and functions of $G$ regular on $\operatorname{co}(G)$. Since differential operators with variable coefficients often can be considered as perturbations of operators with constant coefficients, the results mentioned above give us estimates for functions and bounds for the spectra of differential operators with variable coefficients.

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“…Note that the considerably more general operator functions on tensor products of spaces have been considered in [20] and references given therein. f (sI n +D) n .…”

Section: Applications To Operator Functionsmentioning

confidence: 99%

Similarity of Operators on Tensor Products of Spaces and Matrix Differential Operators

GIL’

2018

J. Aust. Math. Soc.

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“…Nevertheless, B(Y, X) can be represented (see example 3(e) below) as an extended tensor product X ⊠ Y * which enables one to treat it almost as a usual tensor product. The exposition in this Section is based on [89].…”

Section: Extended Tensor Productsmentioning

confidence: 99%

“…Let X and Y be Banach spaces. We call an extended tensor product [89] of X and Y a collection consisting of three objects: a Banach space X ⊠ Y (which we briefly refer to as the extended tensor product) and two (not necessarily closed) full unital subalgebras B 0 (X) and B 0 (Y ) of the algebras B(X) and B(Y ) respectively that satisfy assumptions (A), (B), and (C) listed below.…”

Section: Extended Tensor Productsmentioning

confidence: 99%

“…Example 3. We recall [89] some examples of extended tensor products. (a) Let α be a cross-norm on an algebraic tensor product X ⊗ Y .…”

Section: Extended Tensor Productsmentioning

confidence: 99%

“…From the point of view of applications, the last example seems to be the most important. Therefore, in all cases, we call the operator that corresponds to p(A, B) a transformator ; this term is conventional [51] for the mappings of the type C → N i,j=0 c ij A i CB j acting on operators C. In order to embrace the last example and some others, our treatment is based on the notion of an extended tensor product (Section 5) proposed in [89].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Analytic functional calculus for two operators

Kurbatov

Kurbatova

Oreshina

2016

Preprint

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Properties of the mappingsλ dλ are discussed; here R 1, (•) and R 2, (•) are pseudo-resolvents, i. e., resolvents of bounded, unbounded, or multivalued linear operators, and f and g are analytic functions. Several applications are considered: a representation of the impulse response of a second order linear differential equation with operator coefficients, a representation of the solution of the Sylvester equation, and an exploration of properties of the differential of the ordinary functional calculus.

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Extended tensor products and an operator-valued spectral mapping theorem

Cited by 2 publications

References 14 publications

Similarity of Operators on Tensor Products of Spaces and Matrix Differential Operators

Similarity of Operators on Tensor Products of Spaces and Matrix Differential Operators

Analytic functional calculus for two operators

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