w-Hyponormal Operators have Scalar Extensions

In this paper we show that every w-hyponormal operator has a scalar extension, i.e. is similar to the restriction to an invariant subspace of a scalar operator of order 4. As a corollary, we obtain that every w-hyponormal operator satisfies the property (β).     

Some connections between an operator and its Helton class

In this paper we show that the nilpotent perturbation of operators in the Helton class of p-hyponormal operators has scalar extensions. As a corollary we get that the nilpotent perturbation of each operator in the Helton class of p-hyponormal operators has a nontrivial invariant subspace if its spectrum has nonempty interior in the plane.     

Invariant subspaces of positive strictly singular operators on Banach lattices

It is shown that every positive strictly singular operator T on a Banach lattice satisfying certain conditions is AM-compact and has invariant subspaces. Moreover, every positive operator commuting with T has an invariant subspace. It is also proved that on such spaces the product of a disjointly strictly singular and a regular AM-compact operator is strictly singular. Finally, we prove that on these spaces the known invariant subspace results for compact-friendly operators can be extended to strictly singular-friendly operators.     

Small Transitive Families of Dense Operator Ranges

In complex, separable, infinite-dimensional Hilbert space there exist 5 proper dense operator ranges with the property that every operator leaving each of them invariant is a scalar multiple of the identity. The algebra of operators leaving a pair of proper dense operator ranges invariant can have an infinite nest of invariant subspaces. A slight extension of Foiaş' Theorem shows that it can also have a non-trivial reducing subspace. Submitted: July 13, 2001? Revised: December 6, 2001.     

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.     

Composition operators with weak hyponormality

There are many operator classes that are weaker than p-hyponormal. These include p-quasihyponormal, absolute p-paranormal, p-paranormal, normaloid, and spectraloid. In this note, we discuss measure theoretic composition operators in these classes.     

Finitely strictly singular operators between James spaces

An operator between Banach spaces is said to be finitely strictly singular if for every ε>0 there exists n such that every subspace E⊆X with dimE?n contains a vector x such that ‖Tx‖<ε‖x‖. We show that, for 1?p<q<∞, the formal inclusion operator from Jp to Jq is finitely strictly singular. As a consequence, we obtain that the strictly singular operator with no invariant subspaces constructed by C. Read is actually finitely strictly singular. These results are deduced from the following fact: if k?n then every k-dimensional subspace of Rn contains a vector x with ‖x?_∞‖=1 such that xmi=i(−1) for some m₁<?<mk.     

Backward Aluthge Iterates of a Hyponormal Operator Have Scalar Extensions

The backward Aluthge iterate (defined below) of a hyponormal operator was initiated in [11]. In this paper we characterize the backward Aluthge iterate of a weighted shift. Also we show that the backward Aluthge iterate of a hyponormal operator has an analogue of the single valued extension property for . Finally, we show that backward Aluthge iterates of a hyponormal operator have scalar extensions. As a corollary, we get that the backward Aluthge iterate of a hyponormal operator has a nontrivial invariant subspace if its spectrum has interior in the plane.     

<Emphasis Type="Italic">K</Emphasis>th Roots of <Emphasis Type="Italic">p</Emphasis>-Hyponormal Operators are Subscalar Operators of Order 4<Emphasis Type="Italic">k</Emphasis>

In this paper, we consider the special case of the question raised by Halmos (see below). In particular, we show that if T^k is p-hyponormal, then T is a subscalar operator of order 4k. As a corollary, we obtain that if T^k is p-hyponormal and σ(T) has nonempty interior in the plane, then T has a nontrivial invariant subspace.     

The L resolvents of elliptic operators with uniformly continuous coefficients

We consider the strongly elliptic operator A of order 2m in the divergence form with bounded measurable coefficients and assume that the coefficients of top order are uniformly continuous. For 1<p<∞, A is a bounded linear operator from the L^p Sobolev space H^m,p into H^−m,p. We will prove that (A−λ)⁻¹ exists in H^−m,p for some λ and estimate its operator norm.     

Standard triangularization of semigroups of non-negative operators

We show that, under mild conditions, a semigroup of non-negative operators on L^p(X,μ) (for 1?p<∞) of the form scalar plus compact is triangularizable via standard subspaces if and only if each operator in the semigroup is individually triangularizable via standard subspaces. Also, in the case of operators of the form identity plus trace class we show that triangularizability via standard subspaces is equivalent to the submultiplicativity of a certain function on the semigroup.     

Maximal invariant subspaces for a class of operators

In this note, we characterize maximal invariant subspaces for a class of operators. Let T be a Fredholm operator and \(1-TT^{*}\in\mathcal{S}_{p}\) for some p≥1. It is shown that if M is an invariant subspace for T such that dim?M ? TM<∞, then every maximal invariant subspace of M is of codimension 1 in M. As an immediate consequence, we obtain that if M is a shift invariant subspace of the Bergman space and dim?M ? zM<∞, then every maximal invariant subspace of M is of codimension 1 in M. We also apply the result to translation operators and their invariant subspaces.     

A counterexample to a conjecture of Matsaev

We present an effective algorithm for estimating the norm of an operator mapping a low-dimensional ?^p space to a Banach space with an easily computable norm. We use that algorithm to show that Matsaev’s proposed extension of the inequality of John von Neumann is false in case p=4. Matsaev conjectured that for every contraction T on L^p (1<p<∞) one has for any polynomial P

‖P(T)‖_L^p→L^p?‖P(S)‖_{?^p(Z⁺)→?^p(Z⁺)}     

k-Quasihyponormal operators are subscalar

In this paper we shall prove that if an operatorT∈L(⊕ ₁ ² H) is an operator matrix of the form $$T = \left( {\begin{array}{*{20}c} {T_1 } & {T_2 } \\ 0 & {T_3 } \\ \end{array} } \right)$$ whereT ₁ is hyponormal andT ₃ ^k =0, thenT is subscalar of order 2(k+1). Hence non-trivial invariant subspaces are known to exist if the spectrum ofT has interior in the plane as a result of a theorem of Eschmeier and Prunaru (see [EP]). As a corollary we get that anyk-quasihyponormal operators are subscalar.     

Almost Invariant Half-Spaces of Operators on Banach Spaces

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 ℓ_p (1 ≤ p < ∞) or c₀.     

Square roots of semihyponormal operators have scalar extensions

In this paper, we study some properties of , i.e., square roots of semihyponormal operators. In particular we show that an operator has a scalar extension, i.e., is similar to the restriction to an invariant subspace of a (generalized) scalar operator (in the sense of Colojoar?-Foia?). As a corollary, we obtain that an operator has a nontrivial invariant subspace if its spectrum has interior in the plane.     

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.     

On the totality of sequences in topological vector spaces

Price and Zink [Ann. of Math.82 (1965), 139–145] gave necessary and sufficient conditions for the existence of a multiplier m so that {mφ_n}₁^∞ is total; that is, the linear span is dense in L²[0, 1], thus answering a question raised by Boas and Pollard [Bull. Amer. Math. Soc.54 (1948), 512–522]. Using techniques similar to those of Price and Zink, it is shown that this result can be extended to more general spaces. Indeed, if X is either a separable Fréchet space or a complete separable p-normed space (0 < p ? 1), then the existence of a continuous linear operator A so that {Aφ_n}₁^∞ spans a dense subspace is implied by the existence of a nested, equicontinuous family of commuting projections which in addition has the properties that the union of their ranges is dense and that, for each projection, the projection of the original sequence is total in the projected space. Conversely, in a Banach space, it is shown that if such an operator exists and is 1-1 and scalar, then such a family of projections also exists. Further, it is shown that the above considerations extend the first half of the Price-Zink result to L^p[0, 1] (0 < p < ∞) and the other half to L^p[0, 1] (1 ? p < ∞).     

Nonpositive curvature in p-Schatten class

We study the geometry of the set Δp, with 1<p<∞, which consists of perturbations of the identity operator by p-Schatten class operators, which are positive and invertible as elements of B(H). These manifolds have natural and invariant Finsler structures. In [C. Conde, Geometric interpolation in p-Schatten class, J. Math. Anal. Appl. 340 (2008) 920-931], we introduced the metric dp and exposed several results about this metric space. The aim of this work is to prove that the space (Δp,dp) behaves in many senses like a nonpositive curvature metric space.     

Ordered K-groups associated to substitutional dynamics

The Matsumoto K₀-group is an interesting invariant of flow equivalence for symbolic dynamical systems. Because of its origin as the K-theory of a certain C^∗-algebra, which is also a flow invariant, this group comes equipped with a flow invariant order structure. We emphasize this order structure and demonstrate how methods from operator algebra and symbolic dynamics combine to allow a computation of it in certain cases, including Sturmian and primitive substitutional shifts. In the latter case we show by example that the ordered group is a strictly finer invariant than the group itself.