Left Invertible Operators Research Articles

Tridiagonal kernels and left-invertible operators with applications to Aluthge transforms

Given scalars a_n (\neq 0) and b_n , n \geq 0 , the tridiagonal kernel or band kernel with bandwidth 1 is the positive definite kernel k on the open unit disc \mathbb{D} defined by k(z, w) = \sum_{n=0}^\infty \big((a_n + b_n z) z^n\big) \big((\bar{a}_n + \bar{b}_n \bar{w}) \bar{w}^n \big) \quad (z, w \in \mathbb{D}). This defines a reproducing kernel Hilbert space \mathcal{H}_k (known as tridiagonal space) of analytic functions on \mathbb{D} with \{(a_n + b_nz) z^n\}_{n=0}^\infty as an orthonormal basis. We consider shift operators M_z on \mathcal{H}_k and prove that M_z is left-invertible if and only if \{|{a_n}/{a_{n+1}}|\}_{n\geq 0} is bounded away from zero. We find that, unlike the case of weighted shifts, Shimorin models for left-invertible operators fail to bring to the foreground the tridiagonal structure of shifts. In fact, the tridiagonal structure of a kernel k , as above, is preserved under Shimorin models if and only if b_0=0 or that M_z is a weighted shift. We prove concrete classification results concerning invariance of tridiagonality of kernels, Shimorin models, and positive operators. We also develop a computational approach to Aluthge transforms of shifts. Curiously, in contrast to direct kernel space techniques, often Shimorin models fail to yield tridiagonal Aluthge transforms of shifts defined on tridiagonal spaces.

Weakly concave operators

We study a class of left-invertible operators which we call weakly concave operators. It includes the class of concave operators and some subclasses of expansive strict $m$-isometries with $m > 2$. We prove a Wold-type decomposition for weakly concave operators. We also obtain a Berger–Shaw-type theorem for analytic finitely cyclic weakly concave operators. The proofs of these results rely heavily on a spectral dichotomy for left-invertible operators. It provides a fairly close relationship, written in terms of the reciprocal automorphism of the Riemann sphere, between the spectra of a left-invertible operator and any of its left inverses. We further place the class of weakly concave operators, as the term $\mathcal {A}_1$, in the chain $\mathcal {A}_0 \subseteq \mathcal {A}_1 \subseteq \ldots \subseteq \mathcal {A}_{\infty }$ of collections of left-invertible operators. We show that most of the aforementioned results can be proved for members of these classes. Subtleties arise depending on whether the index $k$ of the class $\mathcal {A}_k$ is finite or not. In particular, a Berger–Shaw-type theorem fails to be true for members of $\mathcal {A}_{\infty }$. This discrepancy is better revealed in the context of $C^*$- and $W^*$-algebras.

The reflexivity of hyperexpansions and their Cauchy dual operators

We discuss the reflexivity of hyperexpansions and their Cauchy dual operators. In particular, we show that any cyclic completely hyperexpansive operator is reflexive. We also establish the reflexivity of the Cauchy dual of an arbitrary $2$-hyperexpansive operator. As a consequence, we deduce the reflexivity of the so-called Bergman-type operator, that is, a left-invertible operator $T$ satisfying the inequality $TT^* + (T^*T)^{-1} \leqslant 2 I_{\mathcal H}.$

Generalized Multipliers for Left-Invertible Operators and Applications

Generalized multipliers for a left-invertible operator T, whose formal Laurent series U_x(z)=sum _{n=1}^infty (P_ET^{n}x)frac{1}{z^n}+sum _{n=0}^infty (P_E{T^{prime *n}}x)z^n, xin mathcal {H} actually represent analytic functions on an annulus or a disc are investigated. We show that they are coefficients of analytic functions and characterize the commutant of some left-invertible operators, which satisfies certain conditions in its terms. In addition, we prove that the set of multiplication operators associated with a weighted shift on a rootless directed tree lies in the closure of polynomials in z and frac{1}{z} of the weighted shift in the topologies of strong and weak operator convergence.

An analytic model for left invertible weighted translation semigroups

M. Embry and A. Lambert initiated the study of a semigroup of operators $\{S_t\}$ indexed by a non-negative real number $t$ and termed it as weighted translation semigroup. The operators $S_t$ are defined on $L^2(\mathbb R_+)$ by using a weight function. The operator $S_t$ can be thought of as a continuous analogue of a weighted shift operator. In this paper, we show that every left invertible operator $S_t$ can be modeled as a multiplication by $z$ on a reproducing kernel Hilbert space $\cal H$ of vector-valued analytic functions on a certain disc centered at the origin and the reproducing kernel associated with $\cal H$ is a diagonal operator. As it turns out that every hyperexpansive weighted translation semigroup is left invertile, the model applies to these semigroups. We also describe the spectral picture for the left invertible weighted translation semigroup. In the process, we point out the similarities and differences between a weighted shift operator and an operator $S_t.$

A Shimorin-type analytic model on an annulus for left-invertible operators and applications

A new analytic model for left-invertible operators, which extends both Shimorin's analytic model for left-invertible and analytic operators and Gellar's model for bilateral weighted shift is introduced and investigated. We show that a left-invertible operator T, which satisfies certain conditions can be modeled as a multiplication operator Mz on a reproducing kernel Hilbert space of vector-valued analytic functions on an annulus or a disc. A similar result for composition operators in ℓ2-spaces is established.

Wold-type decomposition for some regular operators

This paper is concerned with Wold-type decomposition for regular operators whose orbits under any vector satisfy some growth conditions. Several results on left invertible operators close to isometries are extended. We also give numerous results on the Moore–Penrose inverse for regular operators in this particular setting.

Linear maps between operator algebras preserving certain spectral functions

Let H be an infinite dimensional complex Hilbert space and let be a surjective linear map on B(H) with (I) I 2 K(H), where K(H) denotes the closed ideal of all compact operators on H. If preserves the set of upper semi-Weyl operators and the set of all normal eigenvalues in both directions, then is an automorphism of the algebra B(H). Also the relation between the linear maps preserving the set of upper semi-Weyl operators and the linear maps preserving the set of left invertible operators is considered.

Characterizations of perturbations of spectra of [formula omitted] upper triangular operator matrices

When A∈B(H) and B∈B(K) are given, we denote by MC the operator acting on the infinite dimensional separable Hilbert space H⊕K of the form MC=(AC0B). In this paper, we first give some necessary and sufficient conditions for MC to be a left invertible operator (an upper semi-Weyl, upper semi-Fredholm) operator for some C∈B(K,H), which extend the corresponding results in Cao et al. (2006) [4], Cao and Meng (2005) [5], Hwang and Lee (2001) [12] and Li and Du (2006) [15]. Then we present some counter-examples.

On operators close to isometries

We introduce and discuss a class of operators, to be referred to as oper- ators close to isometries. The Bergman-type operators, 2-hyperexpansions, expansive p- isometries, and certain alternating hyperexpansions are main examples of such operators. We establish a few decomposition theorems for operators close to isometries. Applications are given to the theory of p-isometries and of hyperexpansive operators. 1. Preliminaries. In this paper, we discuss the following fundamental problems from single-variable operator theory. If S inB(H) is a completely non-unitary left-invertible operator, under what conditions does

The intersection of left (right) spectra of 2 × 2 upper triangular operator matrices

When A ∈ B ( H ) and B ∈ B ( K ) are given, we denote by M C the operator matrix acting on the infinite-dimensional separable Hilbert space H ⊕ K of the form M C = A C 0 B . In this paper, for given A and B, the sets ⋂ C ∈ B l ( K , H ) σ l ( M C ) , ⋂ C ∈ Inv ( K , H ) σ l ( M C ) and ⋃ C ∈ Inv ( K , H ) σ l ( M C ) are determined, where σ l ( T ) , B l ( K , H ) and Inv ( K , H ) denote, respectively, the left spectrum of an operator T, the set of all the left invertible operators and the set of all the invertible operators from K into H .

The intersection of essential approximate point spectra of operator matrices

When A ∈ B ( H ) and B ∈ B ( K ) are given, we denote by M C the operator acting on the infinite-dimensional separable Hilbert space H ⊕ K of the form M C = ( A C 0 B ) . In this paper, it is shown that there exists some operator C ∈ B ( K , H ) such that M C is upper semi-Fredholm and ind ( M C ) ⩽ 0 if and only if there exists some left invertible operator C ∈ B ( K , H ) such that M C is upper semi-Fredholm and ind ( M C ) ⩽ 0 . A necessary and sufficient condition for M C to be upper semi-Fredholm and ind ( M C ) ⩽ 0 for some C ∈ Inv ( K , H ) is given, where Inv ( K , H ) denotes the set of all the invertible operators of B ( K , H ) . In addition, we give a necessary and sufficient condition for M C to be upper semi-Fredholm and ind ( M C ) ⩽ 0 for all C ∈ Inv ( K , H ) .

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.

INTERPOLATION PROBLEMS INDUCED BY RIGHT AND LEFT INVERTIBLE OPERATORS AND ITS APPLICATIONS TO SINGULAR INTEGRAL EQUATIONS

INTERPOLATION PROBLEMS INDUCED BY RIGHT AND LEFT INVERTIBLE OPERATORS AND ITS APPLICATIONS TO SINGULAR INTEGRAL EQUATIONS

Operators with left inverses similar to their adjoints

The primary object of this paper is to show that if T is a left invertible operator with a left inverse T1 and if there exists an operator S such that T* =S-'TS and 0 0 cl(W(S)), then T is similar to an isometry. An operator T (a bounded linear transformation) on a Hilbert space H is called an isometry if T*T=I, that is, if there exists a left inverse T1 of T such that T* = T1. A unitary operator is called cramped if and only if its spectrum is contained in an arc of the unit circle with central angle less than T a(T), 0oo(T), a1(T), cl(W(T)) and r(T) will denote the spectrum, the set of isolated points of a(T) that are eigenvalues of finite multiplicity, the left essential spectrum, the closure of the numerical range, and the spectral radius of T, respectively. Before stating results we remark that these results are similar to those obtained by Williams [4]. In [3] U. N. Singh and Kanta Mangla proved that if T is an invertible operator and if there exist an operator S such that T*=S-IT-'S and O 0 cl(W(S)), then T is similar to a unitary operator. This leads to the following generalization. THEOREM 1. If T is a left invertible operator with a left inverse T1 and if there exists an operator S such that T*=S-'T1S and 0 cl(W(S)) then T is similar to an isometry. PROOF. Since 0 0 cl(W(S)), 0 can be separated from cl(W(S)) by a half plane. If necessary, we may replace S by SeO for suitable 0, so that this half plane lies strictly on the right of the imaginary axis. Let A = (S+S*/2); then A is positive and invertible. Let A1l2 denote the positive square root of A. Then A1/2 is invertible. Now T1AT* = J(T1ST* + T1S*T*) = i(ST*T* + T1TS*) = i(S + S*) = A. Received by the editors July 17, 1972 and, in revised form, October 11, 1972 and February 9, 1973. AMS (MOS) subject classifications (1970). Primary 47A 10, 47B 10.