Similarity Orbit Research Articles

Similarity and approximation of EP operators

A Hilbert space operator T is called an EP operator if its range R(T) is closed and R(T)=R(T⁎). First, we show that the class of invertible operators and the class of EP operators have the same norm closure and the same interior. Second, we show that the similarity orbit of an nonzero operator T is contained in the class of EP operators if and only if T is invertible. Finally, we describe when Toeplitz operators and weighted shifts possess the EP property. We obtain similar results for an operator property weaker than the EP property.

Geometric similarity invariants of Cowen-Douglas operators

In 1978, M. J. Cowen and R.G. Douglas introduced a class of operators B_n(\Omega) (known as Cowen-Douglas class of operators) and associated a Hermitian holomorphic vector bundle to such an operator. They gave a complete set of unitary invariants in terms of the curvature and its covariant derivatives. At the same time they asked whether one can use geometric ideas to find a complete set of similarity invariants of Cowen-Douglas operators. We give a partial answer to this question. In this paper, we show that the curvature and the second fundamental form completely determine the similarity orbit of a norm dense class of Cowen-Douglas operators. As an application we show that uncountably many (non-similar) strongly irreducible operators in B_n(\mathbb{D}) can be constructed from a given operator in B_1(\mathbb{D}) . We also characterize a class of strongly irreducible weakly homogeneous operators in B_n(\mathbb{D}) .

Weak similarity orbit of (log)‐self‐similar Markov semigroups on the Euclidean space

AbstractWe start by identifying a class of pseudo‐differential operators, generated by the set of continuous negative definite functions, that are in the weak similarity (WS) orbit of the self‐adjoint log‐Bessel operator on the Euclidean space. These WS relations turn out to be useful to first characterize a core for each operator in this class, which enables us to show that they generate a class, denoted by , of non‐self‐adjoint ‐contraction positive semigroups. Up to a homeomorphism, includes, as fundamental objects in probability theory, the family of self‐similar Markov semigroups on . Relying on the WS orbit, we characterize the nature of the spectrum of each element in that is used in their spectral representation which depends on analytical properties of the Bernstein‐gamma functions defined from the associated negative definite functions, and, it is either the point, residual, approximate or continuous spectrum. We proceed by providing a spectral representation of each element in which is expressed in terms of Fourier multiplier operators and valid, at least, on a dense domain of a natural weighted ‐space. Surprisingly, the domain is the full Hilbert space when the spectrum is the residual one, something which seems to be noticed for the first time in the literature. We end up the paper by presenting a series of examples for which all spectral components are computed explicitly in terms of special functions or recently introduced power series.

Analysis of non-reversible Markov chains via similarity orbits

AbstractIn this paper we develop an in-depth analysis of non-reversible Markov chains on denumerable state space from a similarity orbit perspective. In particular, we study the class of Markov chains whose transition kernel is in the similarity orbit of a normal transition kernel, such as that of birth–death chains or reversible Markov chains. We start by identifying a set of sufficient conditions for a Markov chain to belong to the similarity orbit of a birth–death chain. As by-products, we obtain a spectral representation in terms of non-self-adjoint resolutions of identity in the sense of Dunford [21] and offer a detailed analysis on the convergence rate, separation cutoff and L2-cutoff of this class of non-reversible Markov chains. We also look into the problem of estimating the integral functionals from discrete observations for this class. In the last part of this paper we investigate a particular similarity orbit of reversible Markov kernels, which we call the pure birth orbit, and analyse various possibly non-reversible variants of classical birth–death processes in this orbit.

Similarity orbits of complex symmetric operators

An operator T on a complex Hilbert space H is said to be complex symmetric if T can be represented as a symmetric matrix relative to some orthonormal basis for H. In this article we explore the stability of complex symmetry under the condition of similarity. It is proved that the similarity orbit of an operator T is included in the class of complex symmetric operators if and only if T is an algebraic operator of degree at most 2.

Closed unitary and similarity orbits of normal operators in purely infinite [formula omitted]-algebras

We will investigate the norm closure of the unitary and similarity orbits of normal operators in unital, simple, purely infinite C⁎-algebras. A simple proof to the classification of when two normal operators are approximately unitarily equivalent in said algebras with trivial K1-group will be given using previously known techniques. Using these techniques some upper and lower bounds for the distance between unitary orbits of normal operators will be obtained. In addition, a complete characterization of when one normal operator is in the closed similarity orbit of another normal operator will be given for unital, simple, purely infinite C⁎-algebras and type III factors with separable predual.

The strong closure of the simultaneous similarity orbit of some pairs of finite rank operators

In this paper we give necessary and sufficient conditions for membership in the strong closure of the simultaneous similarity orbit for a class of pairs of finite rank operators.

Common hypercyclic vectors for the unitary orbit of a hypercyclic operator

For a separable, infinite dimensional Hilbert space, it was recently shown by the authors that the similarity orbit of a hypercyclic operator contains a path of operators which is dense in the operator algebra with the strong operator topology, and yet the set of common hypercyclic vectors for the entire path is a dense G δ set. Motivated by that result, we show in the present paper that the unitary orbit of any hypercyclic operator contains a path of operators whose closure contains the entire unitary orbit with the strong operator topology, and yet every nonzero vector in the linear span of the orbit of a given hypercyclic vector is a common hypercyclic vector for the entire path.

Similarity orbits in the Calkin algebra

Let B(H) be the algebra of all bounded linear operators on a complex separable infinite dimensional Hilbert space H. Denote by π the quotient map of B(H) onto the Calkin algebra A(H). In 1984, Apostol et al. raised the following conjecture: If an operator T on H is not similar to a compact perturbation of a Jordan operator, then the similarity orbit of π(T) in A(H) coincides with the π-image of the similarity orbit of T. In this paper, we investigate the structure of similarity orbits in the Calkin algebra and give a negative answer to the above conjecture.

The sum of unitary similarity orbits containing only special operators

Let B ( H ) be the algebra of bounded linear operator acting on a Hilbert space H (over the complex or real field). Characterization is given to A 1 , … , A k ∈ B ( H ) such that for any unitary operators U 1 , … , U k , ∑ j = 1 k U j ∗ A j U j is always in a special class S of operators such as normal operators, self-adjoint operators, unitary operators. As corollaries, characterizations are given to A ∈ B ( H ) such that complex, real or nonnegative linear combinations of operators in its unitary orbit U ( A ) = { U ∗ AU : U unitary } always lie in S .

On the strong closure of the simultaneous similarity orbit of a pair of finite rank operators

In this paper, we give a necessary condition for membership in the strong closure of the simultaneous similarity orbit of a pair of finite rank operators.

Holomorphic geometric models for representations of [formula omitted]-algebras

Representations of Banach–Lie groups are realized on Hilbert spaces formed by sections of holomorphic homogeneous vector bundles. These sections are obtained by means of a new notion of reproducing kernel, which is suitable for dealing with involutive diffeomorphisms defined on the base spaces of the bundles. The theory involves considering complexifications of homogeneous spaces acted on by groups of unitaries, and applies in particular to representations of C ∗ -algebras endowed with conditional expectations. In this way, we present holomorphic geometric models for the Stinespring dilations of completely positive maps. The general results are further illustrated by a discussion of several specific topics, including similarity orbits of representations of amenable Banach algebras, similarity orbits of conditional expectations, geometric models of representations of Cuntz algebras, the relationship to endomorphisms of B ( H ) , and non-commutative stochastic analysis.

Eigenvalues of the Sum of Matrices from Unitary Similarity Orbits

Let A and B be $n\times n$ complex matrices. Characterization is given for the set ${\cal E}(A,B)$ of eigenvalues of matrices of the form $U^*AU+V^*BV$ for some unitary matrices U and V. Consequences of the results are discussed and computer algorithms and programs are designed to generate the set ${\cal E}(A,B)$. The results refine those of Wielandt on normal matrices. Extensions of the results to the sum of matrices from three or more unitary similarity orbits are also considered.

Convergence of the iterated Aluthge transform sequence for diagonalizable matrices

Given an r × r complex matrix T, if T = U | T | is the polar decomposition of T, then, the Aluthge transform is defined by Δ ( T ) = | T | 1 / 2 U | T | 1 / 2 . Let Δ n ( T ) denote the n-times iterated Aluthge transform of T, i.e. Δ 0 ( T ) = T and Δ n ( T ) = Δ ( Δ n − 1 ( T ) ) , n ∈ N . We prove that the sequence { Δ n ( T ) } n ∈ N converges for every r × r diagonalizable matrix T. We show that the limit Δ ∞ ( ⋅ ) is a map of class C ∞ on the similarity orbit of a diagonalizable matrix, and on the (open and dense) set of r × r matrices with r different eigenvalues.

Amenability, Completely Bounded projections, Dynamical Systems and Smooth Orbits

We describe a general method to construct completely bounded idempotent mappings on operator spaces, starting from amenable semigroups of completely bounded mappings. We then explore several applications of that method to injective operator spaces, fixed points of completely contractive mappings, Toeplitz operators, dynamical systems and similarity orbits of group representations.

Completely rank-nonincreasing linear maps on spaces of operators*1

We discuss some results and conjectures concerning rank-nonincreasing, rank-preserving, completely rank-preserving and completely rank-nonincreasing linear maps on spaces of operators. We show that the main conjectures are equivalent to a statement about closures of joint similarity orbits of k -tuples of matrices.

Completely rank-nonincreasing linear maps on spaces of operators

We discuss some results and conjectures concerning rank-nonincreasing, rank-preserving, completely rank-preserving and completely rank-nonincreasing linear maps on spaces of operators. We show that the main conjectures are equivalent to a statement about closures of joint similarity orbits of k-tuples of matrices.

Semigroups Generated by Similarity Orbits of Semi-invertible Operators and Unitary Orbits of Isometries

It is proved that for a left invertible operator A, which is not a sum of a scalar and a compact operator, the semigroup generated by similarity orbit of A contains all left invertible operators with suitable (large enough) deficiency. A corresponding result holds for right invertible operators. For isometries, the same statement is proved for unitary orbits.

H-Unitary and Lorentz Matrices: A Review

Many properties of H-unitary and Lorentz matrices are derived using elementary methods. Complex matrices that are unitary with respect to the indefinite inner product induced by an invertible Hermitian matrix H are called H-unitary, and real matrices that are orthogonal with respect to the indefinite inner product induced by an invertible real symmetric matrix are called Lorentz. The focus is on the analogues of singular value and CS (cos -- sin) decompositions for general H-unitary and Lorentz matrices, and on the analogues of Jordan form, in a suitable basis with certain orthonormality properties, for diagonalizable H-unitary and Lorentz matrices. Several applications are given, including connected components of Lorentz similarity orbits, products of matrices that are simultaneously positive definite and H-unitary, products of reflections, and stability and robust stability.

Homogeneous quasi-invariant subspaces of the fock space

AbstractIn this paper, we prove that two homogeneous quasi-invariant subspaces are similar only if they are equal. Moreover, we exhibit an example to show how to determine the similarity orbits of quasi-invariant subspaces.