Martin Schechter

Semigroups of operators and measures of noncompactness

Abstract It is observed that the perturbation class of an open semigroup in a Banach algebra is a closed two-sided ideal. Certain seminorms on the algebra of bounded operators are introduced; these seminorms induce norms on the quotient algebra modulo the ideal of compact operators. Using these seminorms and an assumption apparently weaker than the metric approximation property it is shown that semiFredholm operators have canonical projections (in the quotient algebra) that are not topological zero divisors. A sufficient condition is found that the converse be true. The special cases of subprojective and superprojective Banach spaces are studied. Some properties of essential spectrum are discussed.

On the essential spectrum of an arbitrary operator. I

For a self-adjoint operator A in Hilbert space, a spectral singularity is said to be in the essential spectrum of A if it is not an isolated eigenvalue of finite multiplicity (cf. Wolf [I]). S ome authors refer to it as a limit point of the spectrum (cf. [2, Section 1331). In 1909 H. Weyl [3] proved that the addition of a symmetric, completely continuous operator to A does not affect the essential spectrum. When A is not self-adjoint (but just assumed to be closed and densely defined in an arbitrary Banach space), there are several possible definitions for the essential spectrum. In analogy with Weyl’s theorem, one would like it to be invariant under arbitrary compact perturbations. The definition given above is not suitable in this respect. On the one hand some operators have point eigenvalues which are not isolated and are carried into the resolvent under a compact perturbation. On the other, there are isolated eigenvalues of finite multiplicity which remain invariant under any compact perturbation. To avoid this difficulty, Wolf [l, 41 changed the requirement that the eigenvalue h be isolated in order not to be in the essential spectrum to the requirement that the range R(A A) of A A be closed and have finite codimension. For a self-adjoint operator the two definitions are equivalent. This definition has the advantage that Weyl’s theorem generalizes, i.e., that an arbitrary compact perturbation leaves the essential spectrum unchanged. J. Schwartz [5] has given a definition equivalent to that of Wolf. The essentia1 spectrum given by this definition will be denotes by ocw(A).

The Fučík Spectrum

In Section 3.5 and again in Section 6.3, we mentioned the situation in which

Unique continuation for Schrodinger operators with unbounded potentials

f\left( {x,t} \right)/t \to {\alpha _ \pm }\left( x \right){\text{ }}as{\text{ }}t \to \pm \infty .

Sign-changing critical points from linking type theorems

(7.1.1)

A bounded mountain pass lemma without the (PS) condition and applications

Let A be a densely defined linear operator in a Banach space X. Wolf [4] defines the essential spectrum of A as the complement of §>A, the set of those complex X for which A —X is closed and (a) a(A —X), the multiplicity of X, is finite, (b) R(A —X), the range of A —X, is closed, (c) (3(A — X), the codimension of R{A — X), is finite. This set, which we denote by z implies that xCzD{B) and Bx — z. I t will be called A-closable if xn-^0t Axn—»0, Bxn—>z implies 2 = 0.

Scattering theory for elliptic operators of arbitrary order

Abstract We consider unique continuation theorems for solution of inequalities ¦Δu(x)¦ ⩽ W(x) ¦u(x)¦ with W allowed to be unbounded. We obtain two kinds of results. One allows W ϵ Lploc( R n) with p ⩾ n − 2 for n > 5, p > 1 3 (2n − 1) for n ⩽ 5 . The other requires fW2 to be −Δ-form bounded for all f ϵ C0∞.

Superlinear elliptic boundary value problems

Preface.-.Critical Points of Functionals.-.Minimax Systems.-.Examples of Minimax Systems.-.Ordinary Differential Equations.-.The Method using Flows.-.Finding Linking Sets.-.Sandwich Pairs.-.Semilinear Problems.-.Superlinear Problems.-.Weak Linking.-.Resonance Problems.-.Rotationally Invariant Solutions.-.Semilinear Wave Equations.-.Type (II) Regions.-.Weak Sandwich Pairs.-.Multiple Solutions.-.Second Order Periodic Systems.-.Bibliography