Alan Lambert

Extended eigenvalues and the volterra operator

In this paper we consider the integral Volterra operator on the space L 2 (0; 1). We say that a complex number is an extended eigenvalue ofV if there exists a nonzero operator X satisfying the equation XV = V X . We show that the set of extended eigenvalues of V is precisely the interval (0;1) and the corresponding eigenvectors may be chosen to be integral operators as well.

L p Multipliers and Nested Sigma-Algebras

Throughout this note (X, F, µ) denotes a complete probability space. All sub sigma algebras of F considered are assumed to be complete with respect to p. We shall explore the relationship between a sigma algebra B ⊂.F and the set of multiplication operators which map L p (X, B, µ) into L p (X,.F, µ). (All vector spaces encountered are with respect to the scalar field ℂ.) These operators are closely related to averaging operators on order ideals in Banach lattices and to operators called conditional expectation-type operators in [1]. However, our primary interest in studying these operators lies in their use in investigating chains of sigma algebras. The next section contains some basic notation and several predominantly measure-theoretic facts frequently referred to in the sequel. Since conditional expectations play a central role in this investigation, a subsection of Section I is devoted to a discussion of these operators.

Subnormal weighted translation semigroups

Abstract A weighted translation semigroup { S t } on L 2 ( R + ) is defined by (S t f)(x) = (φ(x) φ(x − t) )f(x − t) for x ⩾ t and 0 otherwise, where φ is a continuous nonzero scalar-valued function on R + . It is shown that { S t } is subnormal if and only if φ 2 is the product of an exponential function and the Laplace-Stieltjes transform of an increasing function of total variation one. A necessary and sufficient condition for similarity of weighted translation semigroups is developed.

MOMENT SEQUENCES AND BACKWARD EXTENSIONS OF SUBNORMAL WEIGHTED SHIFTS

In this note we examine the relationships between a subnormal shift, the measure its moment sequence generates, and those of a large family of weighted shifts associated with the original shift. We examine the effects on subnormality of adding a new weight or changing a weight. We also obtain formulas for evaluating point mass at the origin for the measure associated with the shift. In addition, we examine the relationship between the measure associated with a subnormal shift and those of a family of shifts substantially different from the original shift.

A HILBERT C*-MODULE VIEW OF SOME SPACES RELATED TO PROBABILISTIC CONDITIONAL EXPECTATION

Abstract Measure theoretic conditional expectation is used to construct certain operator spaces. The conditional expectation induces a Hilbert module structure on these spaces. Application is made of Kasparovs theorem to examine certain multiplier and compact operator algebras associated with the module structure. This examination is related to the now classic results on characterizations of conditional expectation type operators.

Backward extensions of subnormal operators

The concept of backward extension for subnormal weighted shifts is generalized to arbitrary subnormal operators. Several differences and similarities in these contexts are explored, with emphasis on the structure of the underlying measures.

Almost stable spectra and limits of similarities

Suppose that {Dn} is a sequence of invertible operators on a Hilbert space, andDnT Dn−1 converges in norm toT0. Recently, H. Bercovici, C. Foias, and A. Tannenbaum have shown that if {Dn±1∶n=1, 2,...} is contained in a finite dimensional subspace of operators, thenT andT0 must have the same spectral radius. Using this result, R. Teodorescu proved that the resolvents ofT andT0 have the same unbounded component. We show that in fact the spectra differ only by certain eigenvalues ofT0, and the spectrum ofT0 is obtained by “filling in holes” in the spectrum ofT; i.e., by adjoining (all, some, or none of the) bounded components of the resolvent ofT to the spectrum ofT.

Invariance of spectrum for representations of *-algebras on Banach spaces

Let K be a Banach space, B a unital C∗-algebra, and π : B → L(K) an injective, unital homomorphism. Suppose that there exists a function γ : K×K → R+ such that, for all k, k1, k2 ∈ K, and all b ∈ B, (a) γ(k, k) = ‖k‖2, (b) γ(k1, k2) ≤ ‖k1‖ ‖k2‖, (c) γ(πbk1, k2) = γ(k1, πb∗k2). Then for all b ∈ B, the spectrum of b in B equals the spectrum of πb as a bounded linear operator on K. If γ satisfies an additional requirement and B is a W∗-algebra, then the Taylor spectrum of a commuting n-tuple b = (b1, . . . , bn) of elements of B equals the Taylor spectrum of the n-tuple πb in the algebra of bounded operators on K. Special cases of these results are (i) if K is a closed subspace of a unital C∗-algebra which contains B as a unital C∗-subalgebra such that BK ⊆ K, and bK = {0} only if b = 0, then for each b ∈ B, the spectrum of b in B is the same as the spectrum of left multiplication by b on K; (ii) if A is a unital C∗-algebra and J is an essential closed left ideal in A, then an element a of A is invertible if and only if left multiplication by a on J is bijective; and (iii) if A is a C∗-algebra, E is a Hilbert A-module, and T is an adjointable module map on E, then the spectrum of T in the C∗-algebra of adjointable operators on E is the same as the spectrum of T as a bounded operator on E. If the algebra of adjointable operators on E is a W∗-algebra, then the Taylor spectrum of a commuting n-tuple of adjointable operators on E is the same relative to the algebra of adjointable operators and relative to the algebra of all bounded operators on E.

Measurable majorants in L 1

Given a probability space ( X , ℱ, μ) and a σ-algebra A ⊂ ℱ, arguably the most powerful tool in gaining information about an ℱ-measurable function f from restricted knowledge of -measurability is that of the conditional expectation E ( f | ); written throughout the remainder of this note. Two properties of conditional expectation that may be exploited to gain information, but which also limit conditional expectations use are the following.

Operator algebras related to measure preserving transformations of finite order

On etudie des operateurs de la forme Tf=Σ i=0 N a i f o τ i , agissant sur f dans L 2 (X,Σ,m) ou chaque a i est une fonction mesurable et τ est une transformation preservant la mesure sur X