A SUPPLEMENT TO MY PAPER ON REAL AND COMPLEX OPERATOR IDEALS

Abstract

We show that the main problem left open in [2] can be solved using the Banach spaces Z^α recently constructed by Kalton [1]. This gives an example of a complex operator ideal that has no real analogue. It thus shows the richer structure of complex operator ideals compared with the real ones.     

CONSTRAINED CONTROLLABILITY IN NON REFLEXIVE BANACH SPACES

Abstract

Throughout the literature on optimal control in Banach spaces, hypotheses like “separable and reflexive” are frequently encountered. In this note we consider one such case, studied by Peichl and Schappacher. Using techniques from Banach space theory and the theory of vector measures, we show how to remove the hypothesis of reflexivity and translate the problem of controls to one about the strong continuity of an adjoint semigroup on the positive real axis.     

Some Open Problems and Conjectures Associated with the Invariant Subspace Problem

There is a subtle difference as far as the invariant subspace problem is concerned for operators acting on real Banach spaces and operators acting on complex Banach spaces. For instance, the classical hyperinvariant subspace theorem of Lomonosov [Funktsional. Anal. nal. i Prilozhen 7(3)(1973), 55–56. (Russian)], while true for complex Banach spaces is false for real Banach spaces. When one starts with a bounded operator on a real Banach space and then considers some “complexification technique” to extend the operator to a complex Banach space, there seems to be no pattern that indicates any connection between the invariant subspaces of the “real” operator and those of its “complexifications.” The purpose of this note is to examine two complexification methods of an operator T acting on a real Banach space and present some questions regarding the invariant subspaces of T and those of its complexifications Mathematics Subject Classification 1991: 47A15, 47C05, 47L20, 46B99 Y.A. Abramovich: 1945–2003 The research of Aliprantis is supported by the NSF Grants EIA-0075506, SES-0128039 and DMI-0122214 and the DOD Grant ACI-0325846     

THE UNIQUENESS OF OPERATIONAL QUANTITIES IN VON NEUMANN ALGEBRAS

Abstract

Since 1970 a number of operational quantities, characteristic of either the semi-Fredholm operators or of some “ideal” of compact-like operators, have been introduced in the theory of bounded operators between Banach spaces and applied successfully to for example perturbation theory. More recently such quantities have been introduced even in the abstract setting of Fredholm theory in a von Neumann algebra relative to some closed two-sided ideal. We show that in this fairly general setting there is only one “reasonable” set of such quantities—a result which in its present form is to the best of our knowledge new even in the case of B(H), the algebra of all bounded operators on a Hilbert space H. We accomplish this by first of all introducing the concept of a (reduced) minimum modulus in the setting of C*-algebras and developing the relevant techniques. In the process we generalise a result of Nikaido [N].     

Extending Independent Sets to Bases and the Axiom of Choice

We show that the both assertions “in every vector space B over a finite element field every subspace V ? B has a complementary subspace S” and “for every family ?? of disjoint odd sized sets there exists a subfamily ?={F_j:j ?ω} with a choice function” together imply the axiom of choice AC. We also show that AC is equivalent to the statement “in every vector space over ? every generating set includes a basis”.     

Frequent hypercyclicity,chaos, and unconditional Schauder decompositions

We prove that if X is any complex separable infinite-dimensional Banach space with an unconditional Schauder decomposition, X supports an operator T which is chaotic and frequently hypercyclic. In contrast with the complex case, we observe that there are real Banach spaces with an unconditional basis which support no chaotic operator.     

AN EXTREME POINT WHICH IS ORTHOGONAL TO THE RANGE OF THE OPERATOR,CONJUGATED TO AN OPERATOR ON l 1

Abstract

In this note we prove that if a linear bounded operator T from l ₁ to a Banach space Y is not an isomorphism, then there exists an element f = (f ₁, f ₂,…) ε l _∞ such that |f _n| = 1 for every n and dist (f, T*Y*) = 1. This result we apply to Sidon sets in the theory of Fourier series.     

Second-order linear differential equations in a Banach space and splitting operators

We consider a second-order linear differential equation whose coefficients are bounded operators acting in a complex Banach space. For this equation with a bounded right-hand side, we study the question on the existence of solutions which are bounded on the whole real axis. An asymptotic behavior of solutions is also explored. The research is conducted under condition that the corresponding “algebraic” operator equation has separated roots or provided that an operator placed in front of the first derivative in the equation has a small norm. In the latter case we apply the method of similar operators, i.e., the operator splitting theorem. To obtain the main results we make use of theorems on the similarity transformation of a second order operator matrix to a block-diagonal matrix.     

Every Banach ideal of polynomials is compatible with an operator ideal

We show that for each Banach ideal of homogeneous polynomials, there exists a (necessarily unique) Banach operator ideal compatible with it. Analogously, we prove that any ideal of n-homogeneous polynomials belongs to a coherent sequence of ideals of k-homogeneous polynomials.     

Every Banach ideal of polynomials is compatible with an operator ideal

We show that for each Banach ideal of homogeneous polynomials, there exists a (necessarily unique) Banach operator ideal compatible with it. Analogously, we prove that any ideal of n-homogeneous polynomials belongs to a coherent sequence of ideals of k-homogeneous polynomials.     

Polynomially Riesz elements

Abstract

A Banach algebra element a ∈ A is said to be “polynomially Riesz”, relative to the homomorphism T : A → B, if there exists a nonzero complex polynomial p(z) such that the image T p(a) ∈ B is quasinilpotent.     

Bilinear forms on exact operator spaces andB(H)⊗B(H)

LetE, F be exact operator spaces (for example subspaces of theC ^*-algebraK(H) of all the compact operators on an infinite dimensional Hilbert spaceH). We study a class of bounded linear mapsu: E →F ^* which we call tracially bounded. In particular, we prove that every completely bounded (in shortc.b.) mapu: E →F ^* factors boundedly through a Hilbert space. This is used to show that the setOS _n of alln-dimensional operator spaces equipped with thec.b. version of the Banach Mazur distance is not separable ifn>2. As an application we whow that there is more than oneC ^*-norm onB (H) ? B (H), or equivalently that $$B(H) \otimes _{\min } B(H) \ne B(H) \otimes _{\max } B(H),$$ which answers a long standing open question. Finally we show that every “maximal” operator space (in the sense of Blecher-Paulsen) is not exact in the infinite dimensional case, and in the finite dimensional case, we give a lower bound for the “exactness constant”. In the final section, we introduce and study a new tensor product forC ^*-albegras and for operator spaces, closely related to the preceding results.     

Monotonicity and complex convexity in Banach lattices

The goal of this article is to study the relations among monotonicity properties of real Banach lattices and the corresponding convexity properties in the complex Banach lattices. We introduce the moduli of monotonicity of Banach lattices. We show that a Banach lattice E is uniformly monotone if and only if its complexification EC is uniformly complex convex. We also prove that a uniformly monotone Banach lattice has finite cotype. In particular, we show that a Banach lattice is of cotype q for some 2?q<∞ if and only if there is an equivalent lattice norm under which it is uniformly monotone and its complexification is q-uniformly PL-convex. We also show that a real Köthe function space E is strictly (respectively uniformly) monotone and a complex Banach space X is strictly (respectively uniformly) complex convex if and only if Köthe-Bochner function space E(X) is strictly (respectively uniformly) complex convex.     

Additive Derivations on Jordan Banach Pairs

Abstract

We show the invariance of “almost all” primitive ideals under additive derivations on a Jordan Banach pair and we extend the well known result of Johnson and Sinclair to the Jordan Banach pairs framework.     

A NOTE ON LEFT STROUGLY NIL AND LEFT STROUGLY NILPOTENT

Abstract

In [2] van der Walt called a left ideal L of a ring A, left strongly nil, if given 1 ε L and k ε K, K a left ideal. there is an n such that (1+k)ⁿ ε K. L is called left strongly nilpotent if for any left ideal K there exists an m such that (L+K)^m ? K. In this paper we will prove that if A is a left artinian ring (not necessarily with unity) then every left strongly nil left ideal is left strongly nilpotent. This result is a generalization of the main theorem of [2].     

A NOTE ON A PAPER BY NG PENG-NUNG AND LEE PENG-YEE

Abstract

In the paper “Convergence in normed Köthe spaces” (J. Singapore National Academy of Science, 4, 146–148 (1975) M.R. 52 # 11568) Ng Peng-Nung and Lee Peng-Yee obtained a convergence result in the general setting of Banach funcation spaces providing conditions in order that pointwise and weak convergence imply norm convergence. They claim this result to be a generalization of a corresponding well known result in the Lebesgue space L₁ (X, u). To substantiate their claim it is necessary to show that the class of Banach function spaces for which their theorem holds is larger than the class of L₁-spaces. This, we shall show, is unfortunately not the case.     

Uncomplemented uniform algebras

Let A be a closed subalgebra of the algebra of all complex-valued continuous functions on a compact space X, and suppose A contains the constant functions and separates points of X; let I be a closed ideal of A such that for some linear multiplicative functional ? on A we have the relation 0 > ∥?∣r∥ > 1 (for the existence of such an ideal it is sufficient that in the maximal ideal space of the algebra A there exists a Gleason part consisting of at least two points). Then the Banach space A^** is not injective [in particular, A is not a complemented subspace of C(X)].     

Local automorphisms of standard operator algebras

Let X be an infinite-dimensional separable real or complex Banach space and A a closed standard operator algebra on X. Then every local automorphism of A is an automorphism. The assumptions of infinite-dimensionality, separability, and closeness are all indispensable.