The Dirac Operator of a Commuting d-Tuple

Given a commuting d-tuple T=(T₁, …, T_d) of otherwise arbitrary operators on a Hilbert space, there is an associated Dirac operator D_T. Significant attributes of the d-tuple are best expressed in terms of D_T, including the Taylor spectrum and the notion of Fredholmness. In fact, all properties of T derive from its Dirac operator. We introduce a general notion of Dirac operator (in dimension d=1, 2, …) that is appropriate for multivariable operator theory. We show that every abstract Dirac operator is associated with a commuting d-tuple, and that two Dirac operators are isomorphic iff their associated operator d-tuples are unitarily equivalent. By relating the curvature invariant introduced in a previous paper to the index of a Dirac operator, we establish a stability result for the curvature invariant for pure d -contractions of finite rank. It is shown that for the subcategory of all such T that are (a) Fredholm and and (b) graded, the curvature invariant K(T) is stable under compact perturbations. We do not know if this stability persists when T is Fredholm but ungraded, although there is concrete evidence that it does.     

Hereditarily polaroid operators, SVEP and Weyl's theorem

A Banach space operator T∈B(X) is hereditarily polaroid, T∈HP, if every part of T is polaroid. HP operators have SVEP. It is proved that if T∈B(X) has SVEP and R∈B(X) is a Riesz operator which commutes with T, then T+R satisfies generalized a-Browder's theorem. If, in particular, R is a quasi-nilpotent operator Q, then both T+Q and T^∗+Q^∗ satisfy generalized a-Browder's theorem; furthermore, if Q is injective, then also T+Q satisfies Weyl's theorem. If A∈B(X) is an algebraic operator which commutes with the polynomially HP operator T, then T+N is polaroid and has SVEP, f(T+N) satisfies generalized Weyl's theorem for every function f which is analytic on a neighbourhood of σ(T+N), and f^∗(T+N) satisfies generalized a-Weyl's theorem for every function f which is analytic on, and constant on no component of, a neighbourhood of σ(T+N).     

Hyponormal operators with rank-two self-commutators

In this paper it is shown that if T∈L(H) satisfies

(i)

T is a pure hyponormal operator;

(ii)

[T^∗,T] is of rank two; and

(iii)

ker[T^∗,T] is invariant for T,

then T is either a subnormal operator or the Putinar's matricial model of rank two. More precisely, if T_|ker[T^∗,T] has a rank-one self-commutator then T is subnormal and if instead T_|ker[T^∗,T] has a rank-two self-commutator then T is either a subnormal operator or the kth minimal partially normal extension, , of a (k+1)-hyponormal operator Tk which has a rank-two self-commutator for any k∈Z₊. Hence, in particular, every weakly subnormal (or 2-hyponormal) operator with a rank-two self-commutator is either a subnormal operator or a finite rank perturbation of a k-hyponormal operator for any k∈Z₊.     

Bifurcation for variational problems when the linearisation has no eigenvalues

For a bounded analytic function, ?, on the unit disk, $\text{D,}\text{let}\text{T}{}_{\mathrm{?}}\text{and}\text{M}{}_{\mathrm{?}}$ denote the operators of multiplication by ? on H²(?D) and L²(?D), respectively. In their 1973 paper, Deddens and Wong asked whether there is an analytic Toeplitz operator $\text{T}{}_{\mathrm{?}}$ that commutes with a nonzero compact operator, and whether every operator that commutes with an analytic Toeplitz operator has an extension that commutes with the corresponding multiplication operator on L². In the first part of this paper, we give an explicit example of an analytic Toeplitz operator T_φ that settles both of these questions. This operator commutes with a nonzero compact operator (a composition operator followed by an analytic Toeplitz operator). The only operators in the commutant of T_φ that extend to commute with M_φ are analytic Toeplitz operators. Although the commutant of T_φ contains more than just analytic Toeplitz operators, T_φ is irreducible. The remainder of the paper seeks to explain more fully the phenomena incorporated in this example by introducing a class of analytic functions, including the function φ, and giving additional conditions on functions g in the class to determine whether T_g commutes with nonzero compact operators, whether T_g is irreducible, and which operators in the commutant of T_g extend to the commutant of M_g. In particular, we find representations for operators in the commutant and second commutant of T_g.     

Constrained von Neumann Inequalities

An equivalent formulation of the von Neumann inequality states that the backward shift S* on ?₂ is extremal, in the sense that if T is a Hilbert space contraction, then ‖p(T)‖?‖p(S*)‖ for each polynomial p. We discuss several results of the following type: if T is a Hilbert space contraction satisfying some constraints, then S* restricted to a suitable invariant subspace is an extremal operator. Several operator radii are used instead of the operator norm. Applications to inequalities of coefficients of rational functions positive on the torus are given.     

Duality properties for b-AM-compact operators on Banach lattices

We obtain some necessary and some sufficient conditions on Banach lattices E and F for the following conditions to hold: (i) if T: E → F is a b-AM-compact operator, then T′: F′ → E′ is also b-AM-compact operator and (ii) if T′: F′ → E′ is b-AM-compact operator, then T: E → F is also b-AM-compact operator.     

Compact operators without extended eigenvalues

A complex number λ is called an extended eigenvalue of a bounded linear operator T on a Banach space B if there exists a non-zero bounded linear operator X acting on B such that XT=λTX. We show that there are compact quasinilpotent operators on a separable Hilbert space, for which the set of extended eigenvalues is the one-point set {1}.     

Lie algebras generated by bounded linear operators on Hilbert spaces

It is proved that the operator Lie algebra ε(T,T^∗) generated by a bounded linear operator T on Hilbert space H is finite-dimensional if and only if T=N+Q, N is a normal operator, [N,Q]=0, and dimA(Q,Q^∗)<+∞, where ε(T,T^∗) denotes the smallest Lie algebra containing T,T^∗, and A(Q,Q^∗) denotes the associative subalgebra of B(H) generated by Q,Q^∗. Moreover, we also give a sufficient and necessary condition for operators to generate finite-dimensional semi-simple Lie algebras. Finally, we prove that if ε(T,T^∗) is an ad-compact E-solvable Lie algebra, then T is a normal operator.     

Invariant subspaces for operators whose spectra are Carathéodory regions

In this paper it is shown that if an operator T satisfies ‖p(T)‖?‖p_‖σ(T) for every polynomial p and the polynomially convex hull of σ(T) is a Carathéodory region whose accessible boundary points lie in rectifiable Jordan arcs on its boundary, then T has a nontrivial invariant subspace. As a corollary, it is also shown that if T is a hyponormal operator and the outer boundary of σ(T) has at most finitely many prime ends corresponding to singular points on ∂D and has a tangent at almost every point on each Jordan arc, then T has a nontrivial invariant subspace.     

Holomorphic functional calculus for operators on a locally convex space

One-variable holomorphic functional calculus is studied on the bornological algebra Lec(E) of all continuous linear oprators on a complete locally convex space E. It is proven that the following three basic notions of the theory are equivalent: (i) existence of projective resolvent of an operator T at a point λ₀, (ii) strict regularity of λ₀ for the operator T in the sense of [12, 13, 15], (iii) tamability of the operator (λ₀ ? T)^?1 (T if λ₀ = ∞), which means that there is a new equivalent system of seminorms on E, such that the operator is bounded in each of them.     

Global weighted inequalities for operators and harmonic forms on manifolds

In this paper, both the local and global weighted Sobolev-Poincaré imbedding inequalities and Poincaré inequalities for the composition T°G are established, where T is the homotopy operator and G is Green's operator applied to A-harmonic forms on manifolds.     

Algebraic extension of *-A operator

In this paper, we study various properties of algebraic extension of *-A operator. Specifically, we show that every algebraic extension of *-A operator has SVEP and is isoloid. And if T is an algebraic extension of *-A operator, then Weyl's theorem holds for f(T), where f is an analytic functions on some neighborhood of σ(T) and not constant on each of the components of its domain.     

Some apparent connection between Baxter and averaging operators

A formula for the resolvent R(λ, T) of a Baxter operator T, on a complex Banach algebra $\text{A}$ with identity e, is obtained. With the parameter θ ≠ 0 and e, but under some restriction, this formula is analogous to that for the resolvent of an averaging operator. A counterexample is given, which shows that such a Baxter operator is not averaging in general. When θ is regular in $\text{A}$, a simple representation of T in terms of summation and averaging operators is obtained.     

Réflexivité d’une extension d’un opérateur normal par un opérateur nilpotent

Let T be an extension of a one to one normal operator A by a nilpotent operator N. In this paper we calculate the defect of reflexivity of T. We give a necessary and sufficient condition to insure the reflexivity of such extensions. In particular, it is shown that T is reflexive when N is reflexive.     

Operators whose dual has non-separable range

Let X and Y be separable Banach spaces and T:X→Y be a bounded linear operator. We characterize the non-separability of T^?(Y^?) by means of fixing properties of the operator T.     

Some aspects of the spectral theory of positive operators

We consider various aspects of the following problem: Let T be a positive operator on a Banach lattice such that σ(T)={1}. Does it follow that T≥1?     

Roots of almost decomposable operators

The author shows that, for an injective analytic function f, f(T) is almost decomposable iff T is almost decomposable, where T is a bounded linear operator on a Banach space and f(T) is defined by the functional calculus.     

More on convexity and smoothness of operators

Let X and Y be Banach spaces and T:Y→X be a bounded operator. In this note, we show first some operator versions of the dual relation between q-convexity and p-smoothness of Banach spaces case. Making use of them, we prove then the main result of this note that the two notions of uniform q-convexity and uniform p-smoothness of an operator T introduced by J. Wenzel are actually equivalent to that the corresponding T-modulus δT of convexity and the T-modulus ρT of smoothness introduced by G. Pisier are of power type q and of power type p, respectively. This is also an operator version of a combination of a Hoffman's theorem and a Figiel-Pisier's theorem. As their application, we show finally that a recent theorem of J. Borwein, A.J. Guirao, P. Hajek and J. Vanderwerff about q-convexity of Banach spaces is again valid for q-convexity of operators.     

Chaotic weighted shifts in Bargmann space

This paper deals with the unilateral backward shift operator T on a Bargmann space F(C). This space can be identified with the sequence space ?²(N). We use the hypercyclicity criterion of Bès, Chan, and Seubert and the program of K.-G. Grosse-Erdmann to give a necessary and sufficient condition in order that T be a chaotic operator. The chaoticity of differentiation which correspond to the annihilation operator in quantum radiation field theory is in view, since the Bargmann space is an infinite-dimensional separable complex Hilbert space.