Spectral Theory of Block Operator Matrices and Applications

Tretter, Christiane

doi:10.1142/p493

Cited by 279 publications

(243 citation statements)

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“…2 and 3, respectively: we refer the interested reader to the monographs [18,47] for more thorough treatments of different aspects of this theory. We are going to study the long-time behavior of semigroups by applying suitable estimates on convolutions of vector-valued functions.…”

Section: ∂ ) (Again (12) Is Obtained Taking the Trace Of The Firstmentioning

confidence: 99%

Asymptotics of semigroups generated by operator matrices

Mugnolo

2014

Arab. J. Math.

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We survey some known results about operator semigroup generated by operator matrices with diagonal or coupled domain. These abstract results are applied to the characterization of well-/ill-posedness for a class of evolution equations with dynamic boundary conditions on domains or metric graphs. In particular, our results on the heat equation with general Wentzell-type boundary conditions complement those previously obtained by, among others, Bandle-von Below-Reichel and Vázquez-Vitillaro.

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Section: ∂ ) (Again (12) Is Obtained Taking the Trace Of The Firstmentioning

confidence: 99%

Asymptotics of semigroups generated by operator matrices

Mugnolo

2014

Arab. J. Math.

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“…The proof involves block operator matrix representations of operators and operator functions relative to various decompositions of the Hilbert space H such as (1.3) and (1.5). If [C 1 ] holds we construct a bounded and boundedly invertible operator function [18], see also [28], and involves the existence of an integral of the resolvent of S. The set of conditions considered in Sect. 4 comes from [24] (and in a weaker form from [11]) and involves the theory of interpolating Hilbert spaces.…”

Section: In Which L + (T) Is An A(t)-invariant J-nonnegative and L − mentioning

confidence: 99%

Conditional Reducibility of Certain Unbounded Nonnegative Hamiltonian Operator Functions

Azizov

Dijksma

Gridneva

2012

Integr. Equ. Oper. Theory

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Abstract. Let J and J be operators on a Hilbert space H which are both self-adjoint and unitary and satisfy JJ = −JJ. We consider an operator function A on [0, 1] of the form A(t) = S + B(t), t ∈ [0, 1], where S is a closed densely defined Hamiltonian (= J-skew-self-adjoint) operator on H with iR ⊂ ρ(S) and B is a function on [0, 1] whose values are bounded operators on H and which is continuous in the uniform operator topology. We assume that for each t ∈ [0, 1] A(t) is a closed densely defined nonnegative (=J-accretive) Hamiltonian operator with iR ⊂ ρ(A(t)). In this paper we give sufficient conditions on S under which A is conditionally reducible, which means that, with respect to a natural decomposition of H, A is diagonalizable in a 2×2 block operator matrix function such that the spectra of the two operator functions on the diagonal are contained in the right and left open half planes of the complex plane. The sufficient conditions involve bounds on the resolvent of S and interpolation of Hilbert spaces.Mathematics Subject Classification. Primary 46C20, 47B50, 47A15; Secondary 47A56, 47B44.

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“…In the sequel, our goal is to replace the tangent lines by cubic curves, introducing an envelope-type set that contains the spectrum and lies in the numerical range. We note in passing that another subset of F (A) that contains the eigenvalues, known as the block numerical range, has been studied extensively; see [9] and [10]. There is no tractable relation known to us between the block numerical range and the envelope.…”

Section: Preliminaries On the Numerical Rangementioning

confidence: 99%

An envelope for the spectrum of a matrix

Psarrakos

Tsatsomeros

2011

Open Mathematics

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We introduce and study an envelope-type region E(A) in the complex plane that contains the eigenvalues of a given n × n complex matrix A. E(A) is the intersection of an infinite number of regions defined by cubic curves. The notion and method of construction of E(A) extend the notion of the numerical range of A, F (A), which is known to be an intersection of an infinite number of half-planes; as a consequence, E(A) is contained in F (A) and represents an improvement in localizing the spectrum of A.

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Spectral Theory of Block Operator Matrices and Applications

Cited by 279 publications

References 0 publications

Asymptotics of semigroups generated by operator matrices

Asymptotics of semigroups generated by operator matrices

Conditional Reducibility of Certain Unbounded Nonnegative Hamiltonian Operator Functions

An envelope for the spectrum of a matrix

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