Minimum modulus, perturbation for essential ascent and descent of a closed linear relation in Hilbert spaces

Z. Garbouj; H. Skhiri

doi:10.1007/s10474-016-0683-1

Minimum modulus, perturbation for essential ascent and descent of a closed linear relation in Hilbert spaces

Acta Mathematica Hungarica ◽

10.1007/s10474-016-0683-1 ◽

2016 ◽

Vol 151 (2) ◽

pp. 328-360

Author(s):

Z. Garbouj ◽

H. Skhiri

Keyword(s):

Hilbert Spaces ◽

Linear Relation ◽

Minimum Modulus ◽

Closed Linear Relation

On the Reduced Minimum Modulus of a Linear Relation in Hilbert Spaces

Complex Analysis and Operator Theory ◽

10.1007/s11785-011-0173-y ◽

2011 ◽

Vol 7 (4) ◽

pp. 801-812 ◽

Cited By ~ 4

Author(s):

Teresa Alvarez ◽

Adrian Sandovici

Keyword(s):

Hilbert Spaces ◽

Linear Relation ◽

Minimum Modulus ◽

Reduced Minimum Modulus

Canonical Graph Contractions of Linear Relations on Hilbert Spaces

Complex Analysis and Operator Theory ◽

10.1007/s11785-020-01066-3 ◽

2021 ◽

Vol 15 (1) ◽

Author(s):

Zsigmond Tarcsay ◽

Zoltán Sebestyén

Keyword(s):

Hilbert Spaces ◽

Linear Relation ◽

Regular Part ◽

Linear Relations ◽

Closed Linear Relation ◽

Canonical Graph

AbstractGiven a closed linear relation T between two Hilbert spaces $$\mathcal {H}$$ H and $$\mathcal {K}$$ K , the corresponding first and second coordinate projections $$P_T$$ P T and $$Q_T$$ Q T are both linear contractions from T to $$\mathcal {H}$$ H , and to $$\mathcal {K}$$ K , respectively. In this paper we investigate the features of these graph contractions. We show among other things that $$P_T^{}P_T^*=(I+T^*T)^{-1}$$ P T P T ∗ = ( I + T ∗ T ) - 1 , and that $$Q_T^{}Q_T^*=I-(I+TT^*)^{-1}$$ Q T Q T ∗ = I - ( I + T T ∗ ) - 1 . The ranges $${\text {ran}}P_T^{*}$$ ran P T ∗ and $${\text {ran}}Q_T^{*}$$ ran Q T ∗ are proved to be closely related to the so called ‘regular part’ of T. The connection of the graph projections to Stone’s decomposition of a closed linear relation is also discussed.

On the Moore--Penrose inverse of a closed linear relation

Publicationes Mathematicae Debrecen ◽

10.5486/pmd.2013.5788 ◽

2014 ◽

Vol 85 (1-2) ◽

pp. 59-72

Author(s):

Teresa Alvarez ◽

Qiaoling Xia

Keyword(s):

Linear Relation ◽

Closed Linear Relation

On the Moore--Penrose inverse of a closed linear relation

Publicationes Mathematicae Debrecen ◽

10.5486/pmd.2014.5788 ◽

2014 ◽

Vol 85 (1-2) ◽

pp. 59-72

Author(s):

TERESA ALVAREZ

Keyword(s):

Linear Relation ◽

Closed Linear Relation

The reduced minimum modulus of Drazin inverses of linear operators on Hilbert spaces

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-06-08377-8 ◽

2006 ◽

Vol 134 (11) ◽

pp. 3309-3317 ◽

Cited By ~ 9

Author(s):

Chun-Yuan Deng ◽

Hong-Ke Du

Keyword(s):

Hilbert Spaces ◽

Linear Operators ◽

Minimum Modulus ◽

Reduced Minimum Modulus

Selfadjoint Extensions of a Closed Linear Relation of Defect One in a Krein Space

Operator Theory and Boundary Eigenvalue Problems ◽

10.1007/978-3-0348-9106-6_12 ◽

1995 ◽

pp. 176-205 ◽

Cited By ~ 2

Author(s):

Peter Jonas ◽

Heinz Langer

Keyword(s):

Linear Relation ◽

Krein Space ◽

Kreĭn Space ◽

Closed Linear Relation ◽

Selfadjoint Extensions

Characteristic Matrix of a Closed Linear Relation

Complex Analysis and Operator Theory ◽

10.1007/s11785-016-0602-z ◽

2016 ◽

Vol 11 (4) ◽

pp. 727-738

Author(s):

Teresa Álvarez

Keyword(s):

Linear Relation ◽

Characteristic Matrix ◽

Closed Linear Relation

Gaussian Hilbert Spaces

10.1017/cbo9780511526169 ◽

1997 ◽

Cited By ~ 257

Author(s):

Svante Janson

Keyword(s):

Hilbert Spaces

On split equality monotone Yosida variational inclusion and fixed point problems for countable infinite families of certain nonlinear mappings in Hilbert spaces

Novi Sad Journal of Mathematics ◽

10.30755/nsjom.10119 ◽

2020 ◽

Vol Accepted ◽

Author(s):

Oluwatosin Temitope Mewomo ◽

Hammed Anuoluwapo Abass ◽

Chinedu Izuchukwu ◽

Olawale Kazeem Oyewole

Keyword(s):

Fixed Point ◽

Hilbert Spaces ◽

Fixed Point Problems ◽

Variational Inclusion ◽

Nonlinear Mappings

On compressions of self-adjoint extensions of a symmetric linear relation with unequal deficiency indices

Ukrainian Mathematical Bulletin ◽

10.37069/1810-3200-2019-16-4-8 ◽

2019 ◽

Vol 16 (4) ◽

pp. 567-587

Author(s):

Vadim Mogilevskii

Keyword(s):

Hilbert Space ◽

Boundary Conditions ◽

Linear Relation ◽

Deficiency Indices ◽

Space Extension ◽

Limit Value ◽

Abstract Boundary ◽

Symmetric Linear Relation ◽

Symmetric Operators

Let $A$ be a symmetric linear relation in the Hilbert space $\gH$ with unequal deficiency indices $n_-A <n_+(A)$. A self-adjoint linear relation $\wt A\supset A$ in some Hilbert space $\wt\gH\supset \gH$ is called an (exit space) extension of $A$. We study the compressions $C (\wt A)=P_\gH\wt A\up\gH$ of extensions $\wt A=\wt A^*$. Our main result is a description of compressions $C (\wt A)$ by means of abstract boundary conditions, which are given in terms of a limit value of the Nevanlinna parameter $\tau(\l)$ from the Krein formula for generalized resolvents. We describe also all extensions $\wt A=\wt A^*$ of $A$ with the maximal symmetric compression $C (\wt A)$ and all extensions $\wt A=\wt A^*$ of the second kind in the sense of M.A. Naimark. These results generalize the recent results by A. Dijksma, H. Langer and the author obtained for symmetric operators $A$ with equal deficiency indices $n_+(A)=n_-(A)$.