In this paper, we define measures of semi-noncompactness in a locally convex topological linear space with respect to a given seminorm, and give some simple properties, including a fixed point theorem for a certain class of condensing mappings.

An inequality of A.S. Markus for the singular values of compact operators is generalized to measurable operators; this inequality is then used as a tool to construct a wide class of rearrangement invariant Banach spaces of measurable operators"

We discuss a non-commutative extension of a well-known result of Calderon characterizing interpolation spaces for the couple $(L^1, L^{\infty})$. The characterization is given in terms of the generalized singular value functions (or decreasing rearrangement) of measurable operators affiliated with a semi-finite von Neumann algebra.

An example is given of a linear transformation which defines a wellbounded operator on $L^p[O, 1] for 1 \leq p \leq \infty$. It is shown that the properties of the decomposition of the identity associated with this operator (and consequently the type of functional calculus that the operator admits) vary markedly depending on the domain space.

Let L be a strongly elliptic partial differential operator of second order, with real coefficients on $L^p(\Omega), 1 \lt p \lt \infty$, with either Dirichlet, or Neumann, or "oblique" boundary conditions. Assume that $\Omega$ is an open, bounded domain with $C^2$ boundary. By adding a oonstant, if necessary, we then establish an $H_\infty$, functional calculus which associates an operator m(L) to each bounded holomorphic function m so that \[ \| m (L) \| \leq M \| m \|\infty \] where M is a constmt independent of m. Under suitable asumptions on L, we can also obtain a similar result in the case of Dirichlet boundary conditions where $\Omega$ is a non-smooth domain.

The recent achievements concerning $m$-curvature equations gave a new point of view to the geometrical theory of mixed volumes of convex bodies which was developed by A.D. Aleksandrov in I938-I940. A principal goal of this paper is to pose corresponding variational problems correctly and to formulate sufficient conditions for the existence of minimizers.

It is shown that if the highest order co-efficients of a uniformly elliptic second order differential operator $L$ on $\mathbb{R}^d$ are bounded and Hölder continuous, and the other coefficients are bounded and measurable, then there is at most one semigroup $S$ acting on bounded Borel measurable functions, such that $S$ is given by a transition function, and for all smooth functons $f$ with compact support in $\mathbb{R}^d, S(t)f(x) + \int_0^t S(s)Lf(x) ds$ for all $t > 0$ and $x \in \mathbb(R)^d$.