Unitary Boundary Pairs for Isometric Operators in Pontryagin Spaces and Generalized Coresolvents

An isometric operator V in a Pontryagin space $${{{\mathfrak {H}}}}$$ H is called standard, if its domain and the range are nondegenerate subspaces in $${{{\mathfrak {H}}}}$$ H . A description of coresolvents for standard isometric operators is known and basic underlying concepts that appear in the literature are unitary colligations and characteristic functions. In the present paper generalized coresolvents of non-standard Pontryagin space isometric operators are described. The methods used in this paper rely on a new general notion of boundary pairs introduced for isometric operators in a Pontryagin space setting. Even in the Hilbert space case this notion generalizes the earlier concept of boundary triples for isometric operators and offers an alternative approach to study operator valued Schur functions without any additional invertibility requirements appearing in the ordinary boundary triple approach.

On n-quasi-$[m,C]$-isometric operators

For positive integers m and n, an operator $T \in B ( H )$T∈B(H) is said to be an n-quasi-$[m,C]$[m,C]-isometric operator if there exists some conjugation C such that T∗n(∑j=0m(−1)j(mj)CTm−jC.Tm−j)Tn=0. In this paper, some basic structural properties of n-quasi-$[m,C]$[m,C]-isometric operators are established with the help of operator matrix representation. As an application, we obtain that a power of an n-quasi-$[m,C]$[m,C]-isometric operator is again an n-quasi-$[m,C]$[m,C]-isometric operator. Moreover, we show that the class of n-quasi-$[m,C]$[m,C]-isometric operators is norm closed. Finally, we examine the stability of n-quasi-$[m,C]$[m,C]-isometric operator under perturbation by nilpotent operators commuting with T.

The properties of [∞,C]-isometric operators

In this paper we introduce the class of [?,C]-isometric operators and study various properties of this class. In particular, we show that if T is an [?,C]-isometric operator and Q is a quasi-nilpotent operator, then T + Q is an [?,C]-isometric operator under suitable conditions. Also, we show that the class of [?,C]-isometric operators is norm closed. Finally, we examine properties of products of [?,C]-isometric operators.

Some properties of (m,C)-isometric operators

In this paper, we study if T is an (m,C)-isometric operator and CT+C commutes with T, then T+ is an (m,C)-isometric operator. We also give local spectral properties and spectral relations of (m;C)-isometric operators, such as property (?), decomposability, the single-valued extension property and Dunford?s boundedness. We also investigate perturbation of (m,C)-isometric operators by nilpotent operators and by algebraic operators and give some properties.

On an elementary operator with 2-isometric operator entries

A Hilbert space operator T is said to be a 2-isometric operator if T*2T2- 2T*T + I = 0. Let dAB ? B(B(H)) denote either the generalized derivation ?AB = LA-RB or the elementary operator ?= LARB-I, we show that if A and B* are 2-isometric operators, then, for all complex ?, (dAB-?)-1(0)? (d*AB-?)-1(0), the ascent of (dAB-?) ? 1, and dis polaroid. Let H(?(dAB)) denote the space of functions which are analytic on ?(dAB), and let Hc(?(dAB)) denote the space of f ? H(?(dAB)) which are non-constant on every connected component of ?(dAB), it is proved that if A and B* are 2-isometric operators, then f(dAB) satisfies the generalized Weyl?s theorem and f(d*AB) satisfies the generalized a-Weyl?s theorem.

On [m,C]-isometric operators

In this paper we introduce an [m;C]-isometric operator T on a complex Hilbert space H and study its spectral properties. We show that if T is an [m,C]-isometric operator and N is an n-nilpotent operator, respectively, then T + N is an [m + 2n ? 2,C]-isometric operator. Finally we give a short proof of Duggal?s result for tensor product of m-isometries and give a similar result for [m,C]-isometric operators.

On n-quasi-m-isometric operators

We introduce the class of [Formula: see text]-quasi-[Formula: see text]-isometric operators on Hilbert space. This generalizes the class of [Formula: see text]-isometric operators on Hilbert space introduced by Agler and Stankus. An operator [Formula: see text] is said to be [Formula: see text]-quasi-[Formula: see text]-isometric if [Formula: see text] In this paper [Formula: see text] matrix representation of a [Formula: see text]-quasi-[Formula: see text]-isometric operator is given. Using this representation we establish some basic properties of this class of operators.

(m, p)-isometric and (m, ∞)-isometric operator tuples on normed spaces

We generalize the notion of m-isometric operator tuples on Hilbert spaces in a natural way to operator tuples on normed spaces. This is done by defining a tuple analogue of (m, p)-isometric operators, so-called (m, p)-isometric operator tuples. We then extend this definition further by introducing (m, ∞)-isometric operator tuples and study properties of and relations between these objects.

The Essential Spectrum of the Essentially Isometric Operator

Let T be a contraction on a complex, separable, infinite dimensional Hilbert space and let σ(T) (resp. σe(T)) be its spectrum (resp. essential spectrum). We assume that T is an essentially isometric operator; that is, IH -T*T is compact. We show that if D\σ(T) ≠ Ø, then for every f from the disc-algebraσe(f(T)) = f(σe(T))where D is the open unit disc. In addition, if T lies in the class C0.∪ C.0, thenσe(f(T)) = f(σ(T) ∩ Γ),where Γ is the unit circle. Some related problems are also discussed.

Supercyclicity and Hypercyclicity of an Isometry Plus a Nilpotent

Suppose thatXis a separable normed space and the operatorsAandQare bounded onX. In this paper, it is shown that ifAQ=QA,Ais an isometry, andQis a nilpotent then the operatorA+Qis neither supercyclic nor weakly hypercyclic. Moreover, if the underlying space is a Hilbert space andAis a co-isometric operator, then we give sufficient conditions under which the operatorA+Qsatisfies the supercyclicity criterion.