A positive operator A and a closed subspace of a Hilbert space ℋ are called compatible if there exists a projector Q onto such that AQ = Q*A. Compatibility is shown to depend on the existence of certain decompositions of ℋ and the ranges of A and $A^{1/2}$. It also depends on a certain angle between A() and the orthogonal of .

Convergence of semigroups which do not converge in the Trotter-Kato-Neveu sense is considered.

Let V ⊂ Z be two subspaces of a Banach space X. We define the set of generalized projections by ${}_V(X,Z):={P\in (X,Z):P|}_V=id$. Now let X = c₀ or $l_{\infty }^m$, Z:= kerf for some f ∈ X* and $V:=Z\cap l{ⁿ}_{\infty }$ (n < m). The main goal of this paper is to discuss existence, uniqueness and strong uniqueness of a minimal generalized projection in this case. Also formulas for the relative generalized projection constant and the strong uniqueness constant will be given (cf. J. Blatter and E. W. Cheney [Ann. Mat. Pura Appl. 101 (1974), 215-227] and G. Lewicki...

We design an abstract setting for the approximation in Banach spaces of operators acting in duality. A typical example are the gradient and divergence operators in Lebesgue-Sobolev spaces on a bounded domain. We apply this abstract setting to the numerical approximation of Leray-Lions type problems, which include in particular linear diffusion. The main interest of the abstract setting is to provide a unified convergence analysis that simultaneously covers (i) all usual boundary conditions, (ii)...

The problem of approximation by accretive elements in a unital C*-algebra suggested by P. R. Halmos is substantially solved. The key idea is the observation that accretive approximation can be regarded as a combination of positive and self-adjoint approximation. The approximation results are proved both in the C*-norm and in another, topologically equivalent norm.

The possibilities of almost sure approximation of unbounded operators in $L_2(X,A,\mu )$ by multiples of projections or unitary operators are examined.