Capacity Theory for Monotone Operators

If Au = - div(a(x, Du)) is a monotone operator defined on the Sobolev space W^1,P(R ⁿ, 1 < p < + , with a(x,0) = 0 for a.e. x Rⁿ, the capacity C_A(E,F) relative to A can be defined for every pair (E,F) of bounded sets in Rⁿ with E F. The main properties of the set function C_A(E,F) are investigated. In particular it is proved that C_A(E,F) is increasing and countably subadditive with respect to E, decreasing with respect to F, and continuous, in a suitable sense, with respect to E and F.     

The essential selfcommutator of a subnormal operator

In this paper an easier proof is obtained of Alexandru Aleman's extension of an inequality of Axler and Shapiro for subnormal operators to the essential norm. The method is applied to show that a hyponormal operator whose essential spectrum has area zero must be essentially normal.

     

The generalized Goursat-Darboux problem for a third order operator

It is proved that if a generalized Goursat-Darboux problem is -well posed then the operator cannot contain derivatives with respect to one of the variables.

     

Radial limit of lacunary Fourier series with coefficients in non-commutative symmetric spaces

Let be a rearrangement invariant space, an arbitrary set and a von Neumann algebra with a semifinite normal faithful trace. It is proved that the associated symmetric space of measurable operators has -RNP if and only if has -RNP extending in this way some previous results by Q. Xu.

     

On quasispectral maximal subspaces of a class of Volterra-type operators

The concept of quasispectral maximal subspaces for quasinilpotent (but not nilpotent) operators was introduced by M. Omladi\v{c} in 1984. As an application a class of quasinilpotent operators on -spaces, close to the Volterra kernel operator, was studied. In the present Banach function space setting we determine all quasispectral maximal subspaces of analogues of such operators and prove that these subspaces are all the invariant bands. An example is given showing that (in general) they are not all the closed, invariant ideals of the operator.

     

Lifting of Kadec-Klee Properties to Symmetric Spaces of Measurable Operators

We show that if is a separable symmetric Banach function space on the positive half-line, then has the Kadec-Klee property (respectively, uniform Kadec-Klee property) for local convergence in measure if and only if, for every semifinite von Neumann algebra , the associated space of -measurable operators has the same property.

     

Incompleteness of the linear span of the positive compact operators

We show that even in the case of a Banach lattice with an order continuous norm (or whose dual has an order continuous norm) the linear span of the positive compact operators on need not be complete under the regular norm.

     

Weak type bounds for a class of rough operators with power weights

In this note we show that and the fractional integral and maximal operators with rough kernel respectively, are bounded operators from to where and

     

A joint spectral characterization of primeness for C-algebras

We prove that a C-algebra is prime iff for every where denotes Taylor spectrum and are the left and right multiplication operators acting on

     

A theorem of the Dore-Venni type for noncommuting operators

A theorem of the Dore-Venni type for the sum of two closed linear operators is proved, where the operators are noncommuting but instead satisfy a certain commutator condition. This result is then applied to obtain optimal regularity results for parabolic evolution equations and evolutionary integral equations which are nonautonomous. The domains of the involved operators may depend on , but is required to satisfy a certain smoothness property. The results are then applied to parabolic partial differential and integro-differential equations.