Spectra of Scalar‐Type Spectral Operators and Schauder Decompositions

Abstract

I t is well known that every non-empty closed subset of the complex plane is the spectrum of a normal operator in a HUBERT space. This result cannot be extended to the case of BANACH spaces. Indeed, it is shown in [17] that the spectrum of a spectral operator in a GR~THENDIECX space with the DUNFORD-PETTIS property is a finite set; the argument is based on the fact that such a BANACH space has no SCHAUDER decomposition. The purpose of this note is to show that, in a locally convex spice, the existence of an unconditional SCHAUDER decomposition ensures the property that every non-empty closed subset of the complex plane is the spectrum of a scalar-type spectral operator (see Theorem 6). As a consequence, a BANACH space has the above property if and only if it has an unconditional SCHAUDER decomposition (see Corollary 7).Throughout this note, let X be a sequentially complete, locally convex &us-DORFP space and X' its dual space. Let L ( X ) denote the space of all continuous linear operators from X into itself. If the space L ( X ) is equipped with the topology of pointwise convergence on X, then it will be denoted by L , ( X ) . The identity operator on the space X will be denoted by I . Let T be a linear operator defined on a subspace of X with values in X. Suppose that 1 is a complex number such that the operator 1I -T is an injection with dense range whose inverse (U -T)-1 defined on the range of 1I -T is extendable to a continuous operator from X into itself. Then, the extension of (11-T)-1 to X will be denoted by R (1.; T). The resolvent equation R(1; T)-R(C; T ) = -( I -[ ) R ( I ; T ) R([; T )holds whenever I and i are complex numbers for which the operators R(A; T ) and R(C; T ) exist.The resolvent set e(T) of a linear operator T from a subspace of X into X consists of those complex numbers 1 such that there is a neighbourhood U of 1 in the

Boolean algebras of projections in (DF)-and (LF)-spaces

“…An underlying principle is to realise the Boolean algebra (whenever possible) as the range of some spectral measure defined on a a-algebra of sets. The well developed theory of vector and projection-valued measures and integration with respect to them can then be invoked; see [3,4,5,6,10,15,17] and the references therein, for example. To make this realisation possible, there are two minimal but essential properties required of the Boolean algebra; it should be at least a-complete as an abstract Boolean algebra and it should be uniformly bounded (that is, equicontinuous) as a family of continuous linear operators.…”

Schauder Decompositions and the Grothendieck and Dunford-Pettis Properties in Köthe Echelon Spaces of Infinite Order

“…If this series is unconditionally convergent in L s (X), then {P n } ∞ n=1 is called an unconditional Schauder decomposition, [24]. It is precisely such decompositions which are associated with (non-trivial) spectral measures; see (the proof of) Proposition 4.3 above and also Lemma 5 and Theorem 6 in [24].…”

Grothendieck spaces with the Dunford–Pettis property

“…Q n Q m = 0 if n = m) satisfying ∞ n=1 Q n = I , with the series converging in L s (X ). If the series is unconditionally convergent in L s (X ), then {P n } ∞ n=1 is called an unconditional Schauder decomposition [30]. Such decompositions are intimately associated with (non-trivial) spectral measures; see (the proof of) [12,Proposition 4.3] and [30,Lemma 5 and Theorem 6].…”