# Showing papers in "Integral Equations and Operator Theory in 2009"

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TL;DR: In this article, the authors considered integral operators defined by positive definite kernels, where X is a metric space endowed with a strictly positive measure, and the decay rates for the eigenvalues of the integral operator were analyzed.

Abstract: We consider integral operators defined by positive definite kernels \(K : X \times X \rightarrow {\mathbb{C}}\), where X is a metric space endowed with a strictly-positive measure. We update upon connections between two concepts of positive definiteness and upgrade on results related to Mercer like kernels. Under smoothness assumptions on K, we present decay rates for the eigenvalues of the integral operator, employing adapted to our purposes multidimensional versions of known techniques used to analyze similar problems in the case where X is an interval. The results cover the case when X is a subset of \({\mathbb{R}}^{m}\) endowed with the induced Lebesgue measure and the case when X is a subset of the sphere Sm endowed with the induced surface Lebesgue measure.

95 citations

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TL;DR: In this paper, the main result is that a particular set of functions related to the controllability of the heat equation with memory and finite signal speed, with a suitable kernel, is a Riesz system.

Abstract: The main result we derive is the proof that a particular set of functions related to the controllability of the heat equation with memory and finite signal speed, with suitable kernel, is a Riesz system Riesz systems are important tools in applied mathematics, for example for the solution of inverse problems In this paper we shows that the Riesz system we identify can be used to give a constructive method for the computation of the control steering a given initial condition to a prescribed target

54 citations

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TL;DR: In this article, the authors investigate properties of particular Banach spaces of Lipschitz functions on a metric space and semigroups defined on their (pre)duals.

Abstract: Interpretation, derivation and application of a variation of constants formula for measure-valued functions motivate our investigation of properties of particular Banach spaces of Lipschitz functions on a metric space and semigroups defined on their (pre)duals. Spaces of measures densely embed into these preduals. The metric space embeds continuously in these preduals, even isometrically in a specific case. Under mild conditions, a semigroup of Lipschitz transformations on the metric space then embeds into a strongly continuous semigroups of positive linear operators on these Banach spaces generated by measures.

49 citations

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TL;DR: In this article, the authors studied the boundedness of operator families under cotype and type assumptions on X and Y and gave sufficient conditions for R$-boundedness of these families.

Abstract: In this paper we study $R$-boundedness of operator families $\mathcal{T}\subset \calL(X,Y)$, where $X$ and $Y$ are Banach spaces. Under cotype and type assumptions on $X$ and $Y$ we give sufficient conditions for $R$-boundedness. In the first part we show that certain integral operator are $R$-bounded. This will be used to obtain $R$-boundedness in the case that $\mathcal{T}$ is the range of an operator-valued function $T:\R^d\to \calL(X,Y)$ which is in a certain Besov space $B^{d/r}_{r,1}(\R^d;\calL(X,Y))$. The results will be applied to obtain $R$-boundedness of semigroups and evolution families, and to obtain sufficient conditions for existence of solutions for stochastic Cauchy problems.

41 citations

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TL;DR: In this paper, it was shown that it is possible to describe explicitly certain spaces such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the Weyl M-function is analytic.

Abstract: In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M-function see the same singularities as the resolvent of a certain restriction AB of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces \({\mathcal{S}}\) and \(\tilde{\mathcal{S}}\) such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M-function is analytic. We present three examples – one involving a Hain-Lust type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary differential operators on a half line – which together indicate that the abstract results are probably best possible.

39 citations

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TL;DR: In this paper, the concepts of compact and finite rank perturbations are defined with the help of the orthogonal projections PA and PB in \({\math{H}} \oplus {\mathcal{K}}\) onto the graphs of A and B.

Abstract: For closed linear operators or relations A and B acting between Hilbert spaces \({\mathcal{H}}\) and \({\mathcal{K}}\) the concepts of compact and finite rank perturbations are defined with the help of the orthogonal projections PA and PB in \({\mathcal{H}} \oplus {\mathcal{K}}\) onto the graphs of A and B. Various equivalent characterizations for such perturbations are proved and it is shown that these notions are a natural generalization of the usual concepts of compact and finite rank perturbations.

35 citations

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TL;DR: In this paper, generalized polar decompositions of densely defined closed linear operators in Hilbert spaces are studied and some applications to relatively (form) bounded and relatively compact perturbations of self-adjoint, normal, and m-sectorial operators are provided.

Abstract: We study generalized polar decompositions of densely defined closed linear operators in Hilbert spaces and provide some applications to relatively (form) bounded and relatively (form) compact perturbations of self-adjoint, normal, and m-sectorial operators.

34 citations

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TL;DR: In this article, it was shown that the Aluthge transform of a complex symmetric operator is unitarily equivalent for s, t > 0, and if T belongs to class wA(t, t), then T is normal.

Abstract: In this paper, we reprove that: (i) the Aluthge transform of a complex symmetric operator \(\tilde{T} = |T|^{\frac{1}{2}} U|T|^{\frac{1}{2}}\) is complex symmetric, (ii) if T is a complex symmetric operator, then \((\tilde{T})^{*}\) and \(\widetilde{T^{*}}\) are unitarily equivalent. And we also prove that: (iii) if T is a complex symmetric operator, then \(\widetilde{(T^{*})}_{s,t}\) and \((\tilde{T}_{t,s})^{*}\) are unitarily equivalent for s, t > 0, (iv) if a complex symmetric operator T belongs to class wA(t, t), then T is normal.

30 citations

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TL;DR: A bounded linear operator T on a Hilbert space H is called an m-isometry for a positive integer m if as discussed by the authors shows that every orbit of T is eventually norm increasing and some misometries can not be N-supercyclic.

Abstract: A bounded linear operator T on a Hilbert space H is called an m-isometry for a positive integer m if $$\sum
olimits_{{k = 0}}^{m} {( - 1)^{{m - k}} } \left( {\begin{array}{*{20}c} m \\ k\\ \end{array} } \right)T^{{*k}} T^{k} = 0$$
We prove some properties concerning the behaviour of the orbit of an m-isometry For example, every orbit of an m-isometry is eventually norm increasing and some m-isometries can not be N-supercyclic, that is, there does not exist an N-dimensional subspace E
N
such that the orbit of T at E
N
is dense in H

30 citations

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TL;DR: In this article, the spectral subspaces of an off-diagonal self-adjoint operator on a Hilbert space were studied under a symmetric perturbation, where the spectral sets σ0 and σ1 are associated with the spectral set σ 1.

Abstract: Let A be a self-adjoint operator on a Hilbert space \(\mathfrak{H}\). Assume that the spectrum of A consists of two disjoint components σ0 and σ1. Let V be a bounded operator on \(\mathfrak{H}\), off-diagonal and J-self-adjoint with respect to the orthogonal decomposition \(\mathfrak{H} = \mathfrak{H}_{0}\, \oplus\, \mathfrak{H}_{1}\) where \(\mathfrak{H}_{0}\) and \(\mathfrak{H}_{1}\) are the spectral subspaces of A associated with the spectral sets σ0 and σ1, respectively. We find (optimal) conditions on V guaranteeing that the perturbed operator L = A + V is similar to a self-adjoint operator. Moreover, we prove a number of (sharp) norm bounds on the variation of the spectral subspaces of A under the perturbation V. Some of the results obtained are reformulated in terms of the Krein space theory. As an example, the quantum harmonic oscillator under a \({\mathcal{PT}}\)-symmetric perturbation is discussed.

29 citations

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TL;DR: In this article, an inverse problem for the Sturm-Liouville equation with an interior discontinuity is considered, and it is shown that the potential function can be uniquely determined by a set of values of eigenfunctions at some interior point and parts of two spectra.

Abstract: In this work, we consider an inverse problem for the Sturm-Liouville equation with an interior discontinuity, and show that the potential function can be uniquely determined by a set of values of eigenfunctions at some interior point and parts of two spectra.

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TL;DR: In this paper, the operational properties of two integral transforms of Fourier type were presented, and the formulation of convolutions for those transforms were derived and applied to linear partial differential equations and an integral equation with mixed Toeplitz-Hankel kernel.

Abstract: In this paper we present the operational properties of two integral transforms of Fourier type, provide the formulation of convolutions, and obtain eight new convolutions for those transforms. Moreover, we consider applications such as the construction of normed ring structures on \(L_{1}({\mathbb{R}})\), further applications to linear partial differential equations and an integral equation with a mixed Toeplitz-Hankel kernel.

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TL;DR: In this paper, the authors discuss algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball in \({\mathbb{C}}^{n}\).

Abstract: In this paper, we discuss some algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball in \({\mathbb{C}}^{n}\). We first determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator. Next, we investigate the zero-product problem for several Toeplitz operators with radial symbols. Also, the corresponding commuting problem of Toeplitz operators whose symbols are of the form \(\xi^{k} \varphi\) is studied, where \(k \in {\mathbb{Z}}^{n}\) and φ is a radial function.

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TL;DR: In this paper, it was shown that Lie derivations of a reflexive algebra on a Banach space are standard if it is a nest, or has the non-trivial smallest element.

Abstract: A Lie derivation is called standard if it is a sum of a derivation and a linear map with image in the center vanishing on commutators. In this paper we show that Lie derivations of a reflexive algebra $${\rm Alg} \mathcal {L}$$
on a Banach space are standard if $$\mathcal {L}$$
is a nest, or has the non-trivial smallest element, or has the non-trivial greatest element.

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TL;DR: In this article, it was shown that the hyponormality of the Toeplitz operator is also a necessary and sufficient condition under certain assumptions, which generalizes the results of I. S. Hwang and J. Lee.

Abstract: Consider \(\varphi = f + \overline {g}\), where f and g are polynomials, and let \(T_{\varphi}\) be the Toeplitz operators with the symbol \(\varphi\). It is known that if \(T_{\varphi}\) is hyponormal then \(|f'(z)|^{2} \geq |g'(z)|^{2}\) on the unit circle in the complex plane. In this paper, we show that it is also a necessary and sufficient condition under certain assumptions. Furthermore, we present some necessary conditions for the hyponormality of \(T_{\varphi}\) on the weighted Bergman space, which generalize the results of I. S. Hwang and J. Lee.

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TL;DR: In this article, a spectral analysis of the normal product of two self-adjoint operators H and K is given, and it is shown that the spectrum of one of the operators is sufficiently asymmetric.

Abstract: We give a spectral analysis of some unbounded normal product HK of two self-adjoint operators H and K (which appeared in [7]) and we say why it is not self-adjoint even if the spectrum of one of the operators is sufficiently “asymmetric”. Then, we investigate the self-adjointness of KH (given it is normal) for arbitrary self-adjoint H and K by giving a counterexample and some positive results and hence finishing off with the whole question of normal products of self-adjoint operators (appearing in [1, 7, 12]).

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TL;DR: In this paper, the authors considered the initial boundary value problem for the time-fractional diffusion equation by using the single layer potential representation for the solution and derived the equivalent boundary integral equation.

Abstract: Here we consider initial boundary value problem for the time–fractional diffusion equation by using the single layer potential representation for the solution. We derive the equivalent boundary integral equation. We will show that the single layer potential admits the usual jump relations and discuss the mapping properties of the single layer operator in the anisotropic Sobolev spaces. Our main theorem is that the single layer operator is coercive in an anisotropic Sobolev space. Based on the coercivity and continuity of the single layer operator we finally show the bijectivity of the operator in a certain range of anisotropic Sobolev spaces.

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TL;DR: In this paper, a generalized Schur class of generalized J-inner mvf's W(λ) which appear as resolvent matrices for bitangential interpolation problems was studied.

Abstract: A class \({\mathcal{U}}_{\kappa 1} (J)\) of generalized J-inner mvf’s (matrix valued functions) W(λ) which appear as resolvent matrices for bitangential interpolation problems in the generalized Schur class of \(p \times q \, {\rm mvf's}\, {\mathcal{S}}_{\kappa}^{p \times q}\) and some associated reproducing kernel Pontryagin spaces are studied. These spaces are used to describe the range of the linear fractional transformation TW based on W and applied to \({\mathcal{S}}_{\kappa 2}^{p \times q}\). Factorization formulas for mvf’s W in a subclass \({\mathcal{U}^{\circ}_{\kappa 1}} (J)\, {\rm of}\, {\mathcal{U}}_{\kappa 1}(J)\) found and then used to parametrize the set \({\mathcal{S}}_{{\kappa 1}+{\kappa 2}}^{p \times q} \cap T_{W} \left[ {\mathcal{S}}_{\kappa 2}^{p \times q} \right]\). Applications to bitangential interpolation problems in the class \({\mathcal{S}}_{{\kappa 1}+{\kappa 2}}^{p \times q}\) will be presented elsewhere.

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TL;DR: In this article, a completeness theorem involving a system of integro-differential equations with some λ-depending boundary conditions is proved, and sufficient conditions for the root functions to form a Riesz basis are established.

Abstract: A completeness theorem is proved involving a system of integro-differential equations with some λ-depending boundary conditions. Also some sufficient conditions for the root functions to form a Riesz basis are established.

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TL;DR: In this paper, the Nevanlinna-Pick problem for a class of subalgebras of H∞ is studied. Butler et al. used a distance formula that generalizes Sarason's [23] work.

Abstract: We study the Nevanlinna-Pick problem for a class of subalgebras of H∞. This class includes algebras of analytic functions on embedded disks, the algebras of finite codimension in H∞ and the algebra of bounded analytic functions on a multiply connected domain. Our approach uses a distance formula that generalizes Sarason’s [23] work. We also investigate the difference between scalar-valued and matrix-valued interpolation through the use of C*-envelopes.

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TL;DR: In this article, the Wick calculus of the Calderon-toeplitz operators is investigated in the context of the admissible wavelet space, and the structure of the space of Calderon wavelets is described.

Abstract: Let G be the “ax + b”-group with the left invariant Haar measure dν and ψ be a fixed real-valued admissible wavelet on $$L_{2}({\mathbb{R}})$$
. The structure of the space of Calderon (wavelet) transforms $$W_{\psi} (L_{2}({\mathbb{R}}))$$
inside $$L_{2}(G, d
u)$$
is described. Using this result some representations, properties and the Wick calculus of the Calderon-Toeplitz operators T
α acting on $$W_{\psi}(L_{2}({\mathbb{R}}))$$
whose symbols a = a(ζ) depend on $$v = \Im\zeta$$
for $$\zeta \in G$$
are investigated.

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TL;DR: In this article, a modified version of the Invariant subspace problem is introduced and studied, where every operator T on an infinite-dimensional Banach space has an almost invariant half-space, that is, a subspace Y of infinite dimension and infinite codimension such that Y is of finite co-occurrence in T(Y).

Abstract: We introduce and study the following modified version of the Invariant Subspace Problem: whether every operator T on an infinite-dimensional Banach space has an almost invariant half-space, that is, a subspace Y of infinite dimension and infinite codimension such that Y is of finite codimension in T(Y). We solve this problem in the affirmative for a large class of operators which includes quasinilpotent weighted shift operators on lp (1 ≤ p < ∞) or c0.

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TL;DR: A Banach space operator T ∈ B(χ) is polaroid if points λ ∈ iso σ(T) are poles of the resolvent of T as discussed by the authors.

Abstract: A Banach space operator T ∈ B(χ) is polaroid if points λ ∈ iso σ(T) are poles of the resolvent of T. Let \(\sigma_a(T), \sigma_w(T), \sigma_{aw}(T), \sigma_{SF_+}(T)\, \rm{and}\,\sigma_{SF_-}(T)\) denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi–Fredholm and lower semi–Fredholm spectrum of T. For A, B and C ∈ B(χ), let MC denote the operator matrix \(\left( {\begin{array}{ll} A & C \\ 0 & B \\ \end{array} } \right)\). If A is polaroid on \(\pi_{0}(M_{C}) = \{{\lambda \in {\rm iso}\, \sigma(M_{C}) : 0 < {\rm dim} (M_{C} - \lambda)^{-1}(0) < \infty}\}\), M0 satisfies Weyl’s theorem, and A and B satisfy either of the hypotheses (i) A has SVEP at points \(\lambda \in \sigma_{w}(M_{0}) \backslash {\sigma_{SF_{+}}}+(A)\) and B has SVEP at points \(\mu \in \sigma_{w}(M_{0}) \backslash {\sigma_{SF_{-}}}(B)\), or, (ii) both A and A* have SVEP at points \(\lambda \in \sigma_{w}(M_{0}) \backslash {\sigma_{SF_{+}}}(A)\), or, (iii) A* has SVEP at points \(\lambda \in \sigma_{w}(M_{0}) \backslash {\sigma_{SF_{+}}}(A)\) and B* has SVEP at points \(\mu \in \sigma _{w}(M_{0}) \backslash \sigma_{SF_{-}}(B)\), then \(\sigma (M_{C}) \backslash \sigma_{w}(M_{C}) = \pi_{0}(M_{C})\). Here the hypothesis that λ ∈ π0(MC) are poles of the resolvent of A can not be replaced by the hypothesis \(\lambda \in \pi_{0}(A)\) are poles of the resolvent of A.

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TL;DR: For any α > −1, let A2α be the weighted Bergman space on the unit ball corresponding to the weight (1 − |z|2)α as discussed by the authors.

Abstract: For any α > −1, let A2α be the weighted Bergman space on the unit ball corresponding to the weight (1 – |z|2)α. We show that if all except possibly one of the Toeplitz operators \(T_{f_{1} },\ldots,T_{f_{r}}\) are diagonal with respect to the standard orthonormal basis of A2α and \(T_{f_{1}} \cdots T_{f_{r}}\) has finite rank, then one of the functions \(f_{1} ,\ldots, f_{r}\) must be the zero function.

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TL;DR: A complete list of homogeneous operators in the Cowen-Douglas class is given in this article, which is obtained from an explicit realization of all the homogeneous Hermitian holomorphic vector bundles on the unit disc under the action of the universal covering group of the bi-holomorphic automorphism group.

Abstract: A complete list of homogeneous operators in the Cowen-Douglas class $$B_n({\mathbb{D}})$$
is given. This classification is obtained from an explicit realization of all the homogeneous Hermitian holomorphic vector bundles on the unit disc under the action of the universal covering group of the bi-holomorphic automorphism group of the unit disc.

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TL;DR: In this article, the authors investigated asymptotic properties of solutions to mixed boundary value problems of thermopiezoelectricity (thermoelectroelasticity) for homogeneous anisotropic solids with interior cracks.

Abstract: We investigate asymptotic properties of solutions to mixed boundary value problems of thermopiezoelectricity (thermoelectroelasticity) for homogeneous anisotropic solids with interior cracks. Using the potential methods and theory of pseudodifferential equations on manifolds with boundary we prove the existence and uniqueness of solutions. The singularities and asymptotic behaviour of the mechanical, thermal and electric fields are analysed near the crack edges and near the curves, where the types of boundary conditions change. In particular, for some important classes of anisotropic media we derive explicit expressions for the corresponding stress singularity exponents and demonstrate their dependence on the material parameters. The questions related to the so called oscillating singularities are treated in detail as well.

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TL;DR: In this paper, an explicit criterion and a Perron-Frobenius type theorem for uniformly asymptotic stability of positive linear Volterra-Stieltjes differential systems are given.

Abstract: We first introduce the notion of positive linear Volterra-Stieltjes differential systems. Then, we give some characterizations of positive systems. An explicit criterion and a Perron-Frobenius type theorem for positive linear Volterra-Stieltjes differential systems are given. Next, we offer a new criterion for uniformly asymptotic stability of positive systems. Finally, we study stability radii of positive linear Volterra-Stieltjes differential systems. It is proved that complex, real and positive stability radius of positive linear Volterra-Stieltjes differential systems under structured perturbations coincide and can be computed by an explicit formula. The obtained results in this paper include ones established recently for positive linear Volterra integro-differential systems [36] and for positive linear functional differential systems [32]-[35] as particular cases. Moreover, to the best of our knowledge, most of them are new.

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TL;DR: In this article, the singular potential of the Bessel operator was derived from its spectrum and the sequence of norming constants, and an algorithm for factorization of non-negative Fredholm operators in the Hilbert space was presented.

Abstract: We study the problem of factorisation of non-negative Fredholm operators acting in the Hilbert space L2(0, 1) and its relation to the inverse spectral problem for Bessel operators. In particular, we derive an algorithm of reconstructing the singular potential of the Bessel operator from its spectrum and the sequence of norming constants.

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TL;DR: It is obtained that a n-quasi-isometry is similar to a power partial isometry if and only if the ranges of the quasinormal partial isometries of Duggal and Aluthge transforms of 2-quasis-isometries are closed.

Abstract: The present paper deals with operators similar to partial isometries. We get some (necessary and) sufficient conditions for the similarity to (adjoints of) quasinormal partial isometries, or more general, to power partial isometries. We illustrate our results on the class of n-quasi-isometries, obtaining that a n-quasi-isometry is similar to a power partial isometry if and only if the ranges \( {\mathcal{R}}(T^j) (1 \leq j \leq n)\) are closed. In particular if n = 2, these conditions ensure the similarity to quasinormal partial isometries of Duggal and Aluthge transforms of 2-quasi-isometries. The case when a n-quasi-isometry is a partial isometry is also studied, and a structure theorem for n-quasi-isometries which are power partial isometries is given.

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TL;DR: In this paper, the authors provided necessary and sufficient conditions for a path of left multiplication operators to have an SOT-dense set of common hypercyclic vectors, and established a natural sufficient condition for such a path to have a common hyper cyclic subspace.

Abstract: An operator on a separable, infinite dimensional Banach space satisfies the Hypercyclicity Criterion if and only if the associated left multiplication operator is hypercyclic; see [14], [16], [29]. By examining paths of operators where each operator along the path satisfies the criterion, we provide necessary and sufficient conditions for a path of left multiplication operators to have an SOT-dense set of common hypercyclic vectors. As a corollary, we establish a natural sufficient condition for a path of operators to have a common hypercyclic subspace.