scholarly journals Matrix measures and finite rank perturbations of self-adjoint operators

2020 ◽  
Vol 10 (4) ◽  
pp. 1173-1210 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil
Keyword(s):  
Finite Rank ◽  
Adjoint Operators ◽  
Matrix Measures

scholarly journals Self adjoint operators and matrix measures

1971 ◽  
Vol 4 (3) ◽  
pp. 289-305 ◽  
Author(s):  
Patrick J. Browne
Keyword(s):  
Hilbert Space ◽  
Representation Theory ◽  
Spectral Representation ◽  
Adjoint Operator ◽  
Multiplication Operator ◽  
Linear Operators ◽  
Positive Matrix ◽  
Adjoint Operators ◽  
Matrix Measures

Given a self adjoint operator, T, on a Hilbert space H, and given an integer n ≥ 1, we produce a collection , N ∈ L, of n × n positive matrix measures and a unitary map U: such that UTU−1, restricted to the co-ordinate space , is the multiplication operator F(t) → tF(t) in that space. This is a generalization of the spectral representation theory of Dunford and Schwartz, Linear operators, II (1966).

Commuting k-Tuples of Self Adjoint Operators and Matrix Measures

1975 ◽  
Vol 18 (3) ◽  
pp. 321-326
Author(s):  
Patrick J. Browne
Keyword(s):  
Hilbert Space ◽  
Representation Theorem ◽  
Spectral Representation ◽  
Adjoint Operators ◽  
Matrix Measures

In this note we give a generalization of a spectral representation theorem for self adjoint operators in a Hilbert space recently obtained by the author [1]. Our interest here is to develop corresponding results for a k-tuple (T1, …, Tk) of commuting self adjoint operators.

scholarly journals Automorphisms and Some Geodesic Properties of Ortho-Grassmann Graphs

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Mark Pankov ◽  
Krzysztof Petelczyc ◽  
Mariusz Źynel
Keyword(s):  
Hilbert Space ◽  
Finite Rank ◽  
Unitary Operator ◽  
Complex Hilbert Space ◽  
Conjugacy Classes ◽  
Grassmann Graph ◽  
Rank 2 ◽  
Adjoint Operators

Let $H$ be a complex Hilbert space. Consider the ortho-Grassmann graph $\Gamma^{\perp}_{k}(H)$ whose vertices are $k$-dimensional subspaces of $H$ (projections of rank $k$) and two subspaces are connected by an edge in this graph if they are compatible and adjacent (the corresponding rank-$k$ projections commute and their difference is an operator of rank $2$). Our main result is the following: if $\dim H\ne 2k$, then every automorphism of $\Gamma^{\perp}_{k}(H)$ is induced by a unitary or anti-unitary operator; if $\dim H=2k\ge 6$, then every automorphism of $\Gamma^{\perp}_{k}(H)$ is induced by a unitary or anti-unitary operator or it is the composition of such an automorphism and the orthocomplementary map. For the case when $\dim H=2k=4$ the statement fails. To prove this statement we compare geodesics of length two in ortho-Grassmann graphs and characterise compatibility (commutativity) in terms of geodesics in Grassmann and ortho-Grassmann graphs. At the end, we extend this result on generalised ortho-Grassmann graphs associated to conjugacy classes of finite-rank self-adjoint operators.

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

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