1 Local Spectral Radius Formulas for a Class of Unbounded Operators on Banach Spaces (original) (raw)

Local spectral radius formulas for a class of unbounded operators on Banach spaces

Journal of Operator Theory, 2013

We exhibit a general class of unbounded operators in Banach spaces which can be shown to have the single-valued extension property, and for which the local spectrum at suitable points can be determined. We show that a local spectral radius formula holds, analogous to that for a globally defined bounded operator on a Banach space with the single-valued extension property. An operator of the class under consideration can occur in practice as (an extension of) a differential operator which, roughly speaking, can be diagonalised on its domain of smooth test functions via a discrete transform, such that the diagonalising transform establishes an isomorphism of topological vector spaces between the domain of the differential operator, in its own topology, and a sequence space. We give concrete examples of (extensions of) such operators (constant coefficient differential operators on the d-torus, Jacobi operators, the Hermite operator, Laguerre operators) and indicate further perspectives.

On the continuous spectrum of a differential operator

Archive for Rational Mechanics and Analysis, 1966

We assume that the potential function q(x) is defined and continuous on G, and bounded over any finite part of G. Let the operator L o be defined by (1) on the space ~(Lo), consisting of all infinitely differentiable complex functions on G which are restrictions to G of functions in C~ ~ (E,), and which vanish on aG. In this paper we formulate a sufficient condition for the operator Lo to be semi-bounded from below, i.e. (L o u, u)>=o~ II u II 2 for some constant ~t and for all ue~(Lo). It then follows from a theorem of WEINHOLTZ-BROWDER that Lo is essentially self-adjoint. We also obtain an estimate for the least point of the continuous spectrum of the self-adjoint extension L. The latter result contains as special cases known results of FRIEDRICHS [5] and RELLICH [9]; it also gives new information in the case G is a "quasi-finite" region and q(x) tends to-oo. Let us select a certain direction in E,; without loss of generality we may suppose it to be the direction of the positive Xt-axis. Let aG [x ~ denote the set

Local spectrum and local spectral radius of an operator at a fixed vector

Studia Mathematica

Let X be a complex Banach space and let L(X) be the Banach algebra of the bounded linear operators on X. The authors characterize continuous linear surjections ϕ:L(X)→L(X) preserving the local spectrum at a fixed vector e≠0. More precisely, given e∈X with e≠0, a continuous linear surjection ϕ:L(X)→L(X) satisfies the equality σ ϕ(T) (e)=σ T (e) between the local spectrum σ ϕ(T) (e) of ϕ(T) at e and the local spectrum σ T (e) of T at e, for every T∈L(X), if and only if there exists an invertible operator A∈L(X) such that Ae=e and ϕ(T)=ATA -1 . Also proved is the following theorem, for e∈X, e≠0: a continuous linear surjection ϕ:L(X)→L(X) satisfies the equality r ϕ(T) (e)=r T (e) between the local spectral radius r ϕ(T) (e) of ϕ(T) at e and the local spectral radius r T (e) of T at e, for every T∈L(X), if and only if there exists a complex number c of modulus 1 and an invertible operator A∈L(X) such that Ae=e and ϕ(T)=cATA -1 .

Eigenfunction expansions for a class of differential operators

Journal of Mathematical Analysis and Applications, 1979

Let Tl be a "perturbation" operator, which we assume to be symmetric (on test-functions) (1.2) Let T = T,, + Tl. We assume that the coefficients bJt) decay at infinity. Corresponding to the rate of this decay, there is a clear distinction between two cases: "Short-range" case, where the bk(t)'s are &-functions (i.e., roughly speaking, j bk(t)l = O(] t I-1-E), E > 0, as 1 t /-f 03) and the "Long-range" case, where, apart from short-range terms, the coefficients are decaying and have short-range derivatives (i.e., roughly, j b,(t)] = O(j t I+) + "short-range", 6 >O, as j t I+co). Under suitable conditions on the coefficients of Tl (to be listed in detail in the next section), T has a self-adjoint "realization" in L, (which is unique only in the case that T is defined on the whole line), which we still denote by T. In a previous paper [2], we have studied some spectral properties of T. The main conclusion obtained can be summarized as follows: In the case that Tl is a longrange perturbation, the essential spectrim of T is absolutely continuous, apart possibly from a sequence of eigenvalues having at most a finite number of limit-points. In this paper, we study the completeness of the eigenfunctions of To and hence derive the unitary equivalence of T,, and the absolutely continuous part of T. Let go be the Fourier transform, (%f) (5) =f(.f) = (27r)-lj2 s" f(x) ecifx dx feLz(R).

Subdivisions of the Spectra for D ( r , 0 , s , 0 , t ) Operator on Certain Sequence Spaces

2019

Let X and Y be Banach spaces and T : X → Y be a bounded linear operator. The set of all bounded linear operators on X into itself is denoted by B(X). The adjoint T ∗ : X → X of T is defined by (T Φ)(x) = Φ(Tx) for all Φ ∈ X and x ∈ X . Clearly, T ∗ is a bounded linear operator on the dual space X. Let T : D(T ) → X a linear operator, defined on D(T ) ⊆ X , where D(T ) denote the domain of T and X is a complex normed linear space. For T ∈ B(X) we associate a complex number α with the operator (T −αI) denoted by Tα defined on the same domain D(T ), where I is the identity operator. The inverse (T − αI), denoted by T α is known as the resolvent operator of T . Many properties of Tα and T α depend on α and spectral theory is concerned with those properties. We are interested in the set of all α in the complex plane such that T α exists. Boundedness of T −1 α is another essential property. We also determine αs for which the domain of T α is dense in X . A regular value is a complex numbe...

Fredholm differential operators with unbounded coefficients

Journal of Differential Equations, 2005

We prove that a first-order linear differential operator G with unbounded operator coefficients is Fredholm on spaces of functions on R with values in a reflexive Banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both R þ and R À and a pair of the ranges of the dichotomy projections is Fredholm, and that the Fredholm index of G is equal to the Fredholm index of the pair. The operator G is the generator of the evolution semigroup associated with the evolution family. In the case when the evolution family is the propagator of a well-posed differential equation u 0 ðtÞ ¼ AðtÞuðtÞ with, generally, unbounded operators AðtÞ; tAR; the operator G is a closure of the operator À d dt þ AðtÞ: Thus, this paper provides a complete infinite-dimensional generalization of well-known finite-dimensional results by Palmer, and by Ben-Artzi and Gohberg. r acting on a space of d-dimensional vector-functions on R; is Fredholm if and only if the differential equation u 0 ðtÞ ¼ AðtÞuðtÞ; tAR; has exponential dichotomies on both R þ ¼ ½0; NÞ and R À ¼ ðÀN; 0; moreover, the Fredholm index of G is equal to the difference of the ranks of the dichotomies. Palmer proved this result in for the case when G acts on a space of continuous vector-functions. Ben-Artzi and Gohberg proved this result in the case when G acts on L 2 ðR; C d Þ and AAL N ðR; LðC d ÞÞ: Also, we remark on an earlier paper by Sacker , where the ''if ''-part of this result and the index formula were proved in the framework of linear skew-product flows over the hull of A: For further developments of the latter approach see , and the bibliographies therein.

Identification of Extrapolation Spaces for Unbounded Operators

Quaestiones Mathematicae, 1996

extrapolation spaces for strongly continuous semigroups of linear operators on Banach spaces have been constructed by various methods (see, e.g., [Am (1988)], ], [Na (1983)], ], [Wa (1986)]). Usually they appear as "artefacts" used in some intermediate step in order to solve the Cauchy problem on the original space. In a few cases (see the papers by the Dutch school on X * , e.g., ]), and in sharp contrast to the situation for interpolation spaces (see, e.g., ], [DiB (1991)], ], ]) the extrapolation spaces have been identified in a concrete way. It is our intention to fill this gap and subsequently to give an application of the extrapolation method to a perturbation problem.

Bounded Point Evaluations and Local Spectral Theory

Eprint Arxiv Math 0008197, 2000

We study in this paper the concept of bounded point evaluations for cyclic operators. We give a negative answer to a question of L.R. Williams {\it Dynamic Systems and Apllications} 3(1994) 103-112. Furthermore, we generalize some results of Williams and give a simple proof of theorem 2.5 of L.R. Williams (The Local Spectra of Pure Quasinormal Operators J. Math. anal. Appl. 187(1994) 842-850) that non normal hyponormal weighted shifts have fat local spectra.