Complementary Schur complements

Bart, Harm; Tsekanovskii, V. E.

doi:10.1016/0024-3795(94)90507-x

Cited by 15 publications

(11 citation statements)

References 4 publications

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“…Hence, only the implication EAE/MC

\Rightarrow

SC remained open. Some confirmative results were obtained in the early 1990s, for matrices , for Hilbert space Fredholm operators and for Banach space Fredholm operators with index 0 . However, the main breakthroughs came in the last five years, most notably, for Hilbert space operators in , initially for the separable case in .…”

Section: Introductionmentioning

confidence: 73%

“…The main feature in these applications is that the relations EAE, MC and SC coincide, and that one can transfer from one to another in a constructive way. This raised the question, posed in , whether EAE, MC and SC may coincide at the level of Banach space operators. In fact, by that time it was known that EAE and MC coincide (see ) and that they are implied by SC (see ), in short, SC

\Rightarrow

EAE

\Leftrightarrow

MC.…”

Section: Introductionmentioning

confidence: 99%

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Equivalence after extension and Schur coupling do not coincide on essentially incomparable Banach spaces

Horst

Messerschmidt

Ran

et al. 2019

Bull. London Math. Soc.

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In 1994, H. Bart and V. É. Tsekanovskii posed the question whether the Banach space operator relations matricial coupling (MC), equivalence after extension (EAE) and Schur coupling (SC) coincide, leaving only the implication EAE/MC ⇒ SC open. Despite several affirmative results, in this paper we show that the answer in general is no. This follows from a complete description of EAE and SC for the case that the operators act on essentially incomparable Banach spaces, which also leads to a new characterisation of the notion of essential incomparability. Concretely, the forward shift operators U on ℓp and V on ℓq, for 1⩽p,q⩽∞, p≠q, are EAE but not SC. As a corollary, SC is not transitive. Under mild assumptions, given U and V that are Atkinson or generalised invertible and EAE, we give a concrete operator W that is SC to both U and V, even if U and V are not SC themselves. Some further affirmative results for the case where the Banach spaces are isomorphic are also obtained.

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“…Hence, only the implication EAE/MC

\Rightarrow

Section: Introductionmentioning

confidence: 73%

\Rightarrow

EAE

\Leftrightarrow

MC.…”

Section: Introductionmentioning

confidence: 99%

Equivalence after extension and Schur coupling do not coincide on essentially incomparable Banach spaces

Horst

Messerschmidt

Ran

et al. 2019

Bull. London Math. Soc.

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show abstract

“…As a consequence, the resolvent operator (L 0 ) −1 of the semi-homogeneous BVP (8.103) implies a representation of the resolvent operator L −1 of the full problem by the help of formula (8.102). For further operator relations, their interplay, and further applications see [6,19,24,50,51,118].…”

Section: From Operator Factorisation To Operator Relationsmentioning

confidence: 99%

From Sommerfeld Diffraction Problems to Operator Factorisation

Speck¹

2019

Constructive Mathematical Analysis

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This article presents a brief survey devoted to the interplay between diffraction and operator theory, particularly between the so-called canonical diffraction problems (exemplified by Sommerfeld half-plane problems) on one hand and operator factorisation theory on the other hand. It is shown how operator factorisation concepts appear in a natural way from applications and how they can help to find solutions rigorously in case of well-posed problems or how to normalise problems by an adequate change of function spaces.

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“…Namely, we have there 20) which is a so-called equivalence after one-sided extension. The equivalence after one-sided extension concept, being stronger than the equivalence after extension, is intimately related with the Schur coupling notion [5,32]. Schur coupled operators allow even more direct relationships between their null spaces and range spaces than in the equivalence after extension relation (cf.…”

Section: Matrical Coupling Let Us Recall Two Definitions and A Few Knmentioning

confidence: 99%

Diffraction from Polygonal-conical Screens, an Operator Approach

Castro

Duduchava

Speck

2014

Operator Theory, Operator Algebras and Applications

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Abstract. The aim of this work is to construct explicitly resolvent operators for a class of boundary value problems in diffraction theory. These are formulated as boundary value problems for the three-dimensional Helmholtz equation with Dirichlet or Neumann conditions on a plane screen of polynomial-conical form (including unbounded and multiplyconnected screens), in weak formulation. The method is based upon operator theoretical techniques in Hilbert spaces, such as the construction of matrical coupling relations and certain orthogonal projections, which represent new techniques in this area of applications. Various cross connections are exposed, particularly considering classical WienerHopf operators in Sobolev spaces as general Wiener-Hopf operators in Hilbert spaces and studying relations between the crucial operators in game. Former results are extended, particularly to multiply-connected screens.Mathematics Subject Classification (2010). Primary 78A45; Secondary 47G30.

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Complementary Schur complements

Cited by 15 publications

References 4 publications

Equivalence after extension and Schur coupling do not coincide on essentially incomparable Banach spaces

Equivalence after extension and Schur coupling do not coincide on essentially incomparable Banach spaces

From Sommerfeld Diffraction Problems to Operator Factorisation

Diffraction from Polygonal-conical Screens, an Operator Approach

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