Classes of Linear Operators Vol. II

Gohberg, Israel; Kaashoek, M. A.; Goldberg, Seymour

doi:10.1007/978-3-0348-8558-4

Cited by 195 publications

(157 citation statements)

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“…(see [8,XXVII.1] or [9]). A straightforward computation involving formulas (3.2) shows that the block operator matrix…”

Section: Lemma 31 Let a ∈ L(x ) Be An N-hypercontraction And Consimentioning

confidence: 99%

Operator-valued Bergman inner functions as transfer functions

Olofsson¹

2008

St. Petersburg Math. J.

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Abstract. An explicit construction characterizing the operator-valued Bergman inner functions is given for a class of vector-valued standard weighted Bergman spaces in the unit disk. These operator-valued Bergman inner functions act as contractive multipliers from the Hardy space into the associated Bergman space, and they have a natural interpretation as transfer functions for a related class of discrete time linear systems. This points to a new interaction between the fields of invariant subspace theory and mathematical systems theory.Let U, X , and Y be general not necessarily separable complex Hilbert spaces, and letbe bounded linear operators. Let n ≥ 1 be an integer. We shall consider operator-valued analytic functions of the formNotice that the function W defined by (0.1) is analytic for z close to the origin and thatThe operator-valued analytic functions of the form (0.1) admit a natural interpretation as transfer functions for a related class of discrete time linear systems. Consider the discrete time linear system generated by the system of recurrence relationsThen the function W given by (0.1) is the transfer function for system (0.2), which means that y(z) = W (z)u(z) (see Theorem 1.1). The notation here is adapted from mathematical systems theory. The space U is the input space, the space Y is the output space, and the space X is the state space of system (0.2). The operator A ∈ L(X ) is called the state space operator, the operator B ∈ L(U, X ) is called the input operator, the operator C ∈ L(X , Y) is called the output operator and the operator D ∈ L(U, Y) is called the feedthrough operator for system (0.2).

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“…(see [8,XXVII.1] or [9]). A straightforward computation involving formulas (3.2) shows that the block operator matrix…”

Section: Lemma 31 Let a ∈ L(x ) Be An N-hypercontraction And Consimentioning

confidence: 99%

Operator-valued Bergman inner functions as transfer functions

Olofsson¹

2008

St. Petersburg Math. J.

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“…[9]). If A(z) also has zero winding number with respect to the origin, the LDU-factors of A and their inverses are in the above weighted Wiener class.…”

Section: Factorization Of Bi-infinite Toeplitz Matricesmentioning

confidence: 99%

“…Necessary and sufficient conditions can be given in terms of the so-called partial indices of .4(z) [14,9], but except in some very specific cases (such as positive definite A(z), rational matrix functions, and matrix polynomials) these indices cannot be easily computed. If A is a positive definite k x k block Toeplitz matrix in the Wiener class, there exists a Wiener-Hopf factorization as in (2.1); in that case D is positive definite and U(z) = ~,(2-1)..…”

Section: J~--oomentioning

confidence: 99%

“…For the examples (1)-(6) of chains given above, one thus obtains expressions for the LDU-factors and their inverses that contain infinite series and infinite products that converge in the strong operator topology (cf. [9]). If the chain is uncountable, the infinite sums and infinite products are to be replaced by integrals and so-called multiplicative integrals, respectively.…”

Section: )mentioning

confidence: 99%

“…The proof for the special case of example (1) given in [9] is easily adapted to examples (2)-(6). THEOREM 8.1.…”

Section: )mentioning

confidence: 99%

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LDU factorization results for bi-infinite and semi-infinite scalar and block Toeplitz matrices

Mee¹,

Rodriguez²,

Seatzu

1996

Calcolo

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ABSTllACT-In this article various existence results for the LDU-factorization of semiinfinite and bi-infinite scalar and block Toeplitz matrices and numerical methods for computing them are reviewed. Moreover, their application to the orthonormalization of splines is indicated. Both banded and non-banded Toeplitz matrices are considered. Extensive use is made of matrix polynomial theory. Results on the approximation by the LDU-factorizations of finite sections are discussed. The generalization of the results to the LDU-factorization of multi-index Toeplitz matrices is outlined.

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Asymptotics of block Toeplitz determinants with piecewise continuous symbols

Basor,

Ehrhardt,

Virtanen

2024

Comm Pure Appl Math

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We determine the asymptotics of the block Toeplitz determinants as for matrix‐valued piecewise continuous functions with a finitely many jumps under mild additional conditions. In particular, we prove that where , , and are constants that depend on the matrix symbol and are described in our main results. Our approach is based on a new localization theorem for Toeplitz determinants, a new method of computing the Fredholm index of Toeplitz operators with piecewise continuous matrix‐valued symbols, and other operator theoretic methods. As an application of our results, we consider piecewise continuous symbols that arise in the study of entanglement entropy in quantum spin chain models.

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Classes of Linear Operators Vol. II

Cited by 195 publications

References 0 publications

Operator-valued Bergman inner functions as transfer functions

Operator-valued Bergman inner functions as transfer functions

LDU factorization results for bi-infinite and semi-infinite scalar and block Toeplitz matrices

Asymptotics of block Toeplitz determinants with piecewise continuous symbols

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