Self-adjoint block operator matrices with non-separated diagonal entries and their Schur complements

H. Langer; A. Markus; V. Matsaev; C. Tretter

doi:10.1016/S0022-1236(02)00115-5

Self-adjoint block operator matrices with non-separated diagonal entries and their Schur complements

H. Langer, A. Markus, V. Matsaev, C. Tretter

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

20 Scopus citations

Abstract

In this paper self-adjoint 2 × 2 block operator matrices A in a Hilbert space ℋ₁ ⊕ ℋ₂ are considered. For an interval which does not intersect the spectrum of at least one of the diagonal entries of A, we prove angular operator representations for the corresponding spectral subspace ℒ(,A) of A and we study the supporting subspace in this angular operator representation ℒ (A), which is the orthogonal projection of ℒ (A) to the corresponding component ℋ₁ or ℋ₂. Our main result is a description of a special direct complement of this supporting subspace in its component in terms of spectral subspaces of the values of the corresponding Schur complement of A in the endpoints of.

Original language	English
Pages (from-to)	427-451
Number of pages	25
Journal	Journal of Functional Analysis
Volume	199
Issue number	2
DOIs	https://doi.org/10.1016/S0022-1236(02)00115-5
State	Published - 20 Apr 2003

Keywords

Angular operator
Block operator matrix
Schur complement

ASJC Scopus subject areas

Analysis

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10.1016/S0022-1236(02)00115-5

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title = "Self-adjoint block operator matrices with non-separated diagonal entries and their Schur complements",

abstract = "In this paper self-adjoint 2 × 2 block operator matrices A in a Hilbert space ℋ1 ⊕ ℋ2 are considered. For an interval which does not intersect the spectrum of at least one of the diagonal entries of A, we prove angular operator representations for the corresponding spectral subspace ℒ(,A) of A and we study the supporting subspace in this angular operator representation ℒ (A), which is the orthogonal projection of ℒ (A) to the corresponding component ℋ1 or ℋ2. Our main result is a description of a special direct complement of this supporting subspace in its component in terms of spectral subspaces of the values of the corresponding Schur complement of A in the endpoints of.",

keywords = "Angular operator, Block operator matrix, Schur complement",

author = "H. Langer and A. Markus and V. Matsaev and C. Tretter",

note = "Funding Information: This work was supported by the Research Training Network HPRN-CT-2000-00116 of the European Union. Funding Information: H. Langer gratefully acknowledges the support of a Dozor Fellowship at the Ben-Gurion University of the Negev. A. Markus thanks the Vienna University of Technology for support and hospitality. V. Matsaev was supported by grant No. 37/00-1 of the Israel Science Foundation. C. Tretter acknowledges the support of the German Research Foundation, DFG, Grant TR 368/4-1, and thanks the Ben-Gurion University of the Negev for support and hospitality.",

year = "2003",

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doi = "10.1016/S0022-1236(02)00115-5",

language = "English",

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journal = "Journal of Functional Analysis",

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TY - JOUR

T1 - Self-adjoint block operator matrices with non-separated diagonal entries and their Schur complements

AU - Langer, H.

AU - Markus, A.

AU - Matsaev, V.

AU - Tretter, C.

N1 - Funding Information: This work was supported by the Research Training Network HPRN-CT-2000-00116 of the European Union. Funding Information: H. Langer gratefully acknowledges the support of a Dozor Fellowship at the Ben-Gurion University of the Negev. A. Markus thanks the Vienna University of Technology for support and hospitality. V. Matsaev was supported by grant No. 37/00-1 of the Israel Science Foundation. C. Tretter acknowledges the support of the German Research Foundation, DFG, Grant TR 368/4-1, and thanks the Ben-Gurion University of the Negev for support and hospitality.

PY - 2003/4/20

Y1 - 2003/4/20

N2 - In this paper self-adjoint 2 × 2 block operator matrices A in a Hilbert space ℋ1 ⊕ ℋ2 are considered. For an interval which does not intersect the spectrum of at least one of the diagonal entries of A, we prove angular operator representations for the corresponding spectral subspace ℒ(,A) of A and we study the supporting subspace in this angular operator representation ℒ (A), which is the orthogonal projection of ℒ (A) to the corresponding component ℋ1 or ℋ2. Our main result is a description of a special direct complement of this supporting subspace in its component in terms of spectral subspaces of the values of the corresponding Schur complement of A in the endpoints of.

AB - In this paper self-adjoint 2 × 2 block operator matrices A in a Hilbert space ℋ1 ⊕ ℋ2 are considered. For an interval which does not intersect the spectrum of at least one of the diagonal entries of A, we prove angular operator representations for the corresponding spectral subspace ℒ(,A) of A and we study the supporting subspace in this angular operator representation ℒ (A), which is the orthogonal projection of ℒ (A) to the corresponding component ℋ1 or ℋ2. Our main result is a description of a special direct complement of this supporting subspace in its component in terms of spectral subspaces of the values of the corresponding Schur complement of A in the endpoints of.

KW - Angular operator

KW - Block operator matrix

KW - Schur complement

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U2 - 10.1016/S0022-1236(02)00115-5

DO - 10.1016/S0022-1236(02)00115-5

M3 - Article

AN - SCOPUS:0038367917

SN - 0022-1236

VL - 199

SP - 427

EP - 451

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 2

ER -

Self-adjoint block operator matrices with non-separated diagonal entries and their Schur complements

Abstract

Keywords

ASJC Scopus subject areas

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