For a metric space X let Cvc(X) (that is, the set of all graphs of real-valued continuous functions with a compact domain in X) be equipped with the Hausdorff metric induced by the hyperspace of nonempty closed subsets of X×R. It is shown that there exists a continuous mapping Φ : Cvc(X) → Cb(X) satisfying the following conditions: (i) Φ(f)| dom f = f for all partial functions f. (ii) For every...

Abstract. Let T1, . . . , Tn be bounded linear operators on a complex Hilbert space H. Then there are compact operators K1, . . . , Kn ∈ B(H) such that the closure of the joint numerical range of the n-tuple (T1−K1, . . . , Tn−Kn) equals to the joint essential numerical range of (T1, . . . , Tn). This generalizes the corresponding result for n = 1. We also show that if S ∈ B(H) and n ∈ N then t...

If the solution of a Lyapunov or Riccati equation corresponding to a finite-dimensional system is invertible, then this leads to additional properties of this system. For infinite-dimensional systems these results do not need to hold. We show that it is necessary to put some additional conditions on the spectrum of the system operator A. For instance, if this operator has compact resolvent, the...

Let (X, d) be a pointed compact metric space, let 0 < α < 1, and let φ : X → X be a base point preserving Lipschitz map. We prove that the essential norm of the composition operator Cφ induced by the symbol φ on the spaces lip0(X, d α) and Lip0(X, d α) is given by the formula ‖Cφ‖e = lim t→0 sup 0<d(x,y)<t d(φ(x), φ(y))α d(x, y)α whenever the dual space lip0(X, d α)∗ has the approximation prope...

ra + n)» T(0x r(£ + v)x F« + n)xk + r(f + t,)x* T(£)xk + T(Qxk T(Qx = T(t + r, h)[T{to)x F(£o)xt] + [r(f + n)** r(ö*J + T({ to)[TQ;0)xt rft,)*] G r({ + t, £0)F2 + F2 + F(£ ¿0)7, C Fi + Vi + Vi C V; i.e., [m+7])-T(0]BCVlor |t?| <Ô, ag£+77go, ÇaF(£). That F(£) is compact for £>£o follows from the fact F(£0) is a compact operator and F(£) ...

Suppose J is a two-sided quasi-Banach ideal of compact operators on a separable innnite-dimensional Hilbert space H. We show that an operator T 2 J can be expressed as nite linear combination of commutators A;B] where A 2 J and B 2 B(H) if and only its eigenvalues (n) (arranged in decreasing order of absolute value, repeated according to algebraic multiplicity and augmented by zeros if necessar...

We study the stability of the reconstruction of the scattering and absorption coefficients in a stationary linear transport equation from knowledge of the full albedo operator in dimension n ≥ 3. The albedo operator is defined as the mapping from the incoming boundary conditions to the outgoing transport solution at the boundary of a compact and convex domain. The uniqueness of the reconstructi...

We consider the problems of preconditioning and iterative inversion of Toeplitz operators on sequences of complex numbers. We divide the preconditioned operator into two parts, of which one is compact and the other is regarded as a small perturbation. It will be shown that the Krylov subspace methods (such as GMRES) will perform initially at superlinear speed when applied to such preconditioned...

Given a complex Banach space X, we investigate the stable character of property (VE) for bounded linear operator T:X→X, under commuting perturbations that are Riesz, compact, algebraic and hereditarily polaroid. We also analyze sufficient conditions allow transfer from tensorial factors T S to its tensor product.

We show that any infinite compact Hausdorff space S is the continuous image of a totally disconnected compact Hausdorff space S', having the same topological weight as S, by a map 95 which admits a regular linear operator of averaging, i.e., a projection of norm one of C(S') onto <p°C(S), where <p°: C(S) -*■ C(S') is the isometric embedding which takes fe C(S) into/« ¡p. A corollary of this the...