Recently, the invertibility of linear combinations of two idempotents has been studied by several authors. Let P and Q be idempotents in a Banach algebra. It was shown that the invertibility of P+ Q is equivalent to that of aP+ bQ for nonzero a, b with a+ b= 0. In this note, we obtain a similar result for square zero operators and those operators satisfying x2= dx for some scalar d. More generally, we show that if P, Q satisfy a quadratic polynomial (x c)(x d) then the linear combination aP+ bQ c (a+ b) being invertible or Fredholm (and the index) is independent of the choice of the nonzero scalars a, b. However, this is not the case when P and Q are involutions, unitaries, partial isometries, k-potents (k 3) and other nilpotents, as counterexamples are provided.

We study the following generalized quasivariational inequality problem: given a closed convex set <i>X</i> in a normed space <i>E</i> with the dual <i>E</i> ^*, a multifunction \Phi :Xightarrow 2^{E^{*}} and a multifunction :<i>X</i>2^<i>X</i>, find a point \Phi :Xightarrow 2^{E^{*}} such that \Phi :Xightarrow 2^{E^{*}} , \Phi :Xightarrow 2^{E^{*}} . We prove some existence theorems in which may be discontinuous, <i>X</i> may be unbounded, and is not assumed to be Hausdorff lower semicontinuous.

We perform quantitative spectral analysis of the self-adjoint Dirichlet Laplacian \mathsf {H} on an unbounded, radially symmetric (generalized) parabolic layer \mathsf {H} . It was known before that \mathsf {H} has an infinite number of eigenvalues below the threshold of its essential spectrum. In the present paper, we find the discrete spectrum asymptotics for \mathsf {H} by means of a consecutive reduction to the analogous asymptotic problem for an effective one-dimensional Schrdinger operator on the half-line with the potential the behaviour of which far away from the origin is determined by the geometry of the layer \mathsf {H} at infinity.

A not necessarily continuous, linear or multiplicative function from an algebra A into itself is called a 2-local automorphism if agrees with an automorphism of A at each pair of points in A. In this paper, we study when a 2-local automorphism of a C-algebra, or a standard operator algebra on a locally convex space, is an automorphism.

Let <i>X, Y</i> be compact Hausdorff spaces and <i>E, F</i> be Banach spaces. A linear map <i>T: C(X, E) C(Y, F)</i> is separating if <i>Tf, Tg</i> have disjoint cozeroes whenever <i>f, g</i> have disjoint cozeroes. We prove that a biseparating linear bijection <i>T</i> (that is, <i>T</i> and <i>T</i>^-1 are separating) is a weighted composition operator <i>Tf = h f</i> o . Here, <i>h</i> is a function from <i>Y</i> into the set of invertible linear operators from <i>E</i> onto <i>F</i>, and , is a homeomorphism from <i>Y</i> onto <i>X</i>. We also show that <i>T</i> is bounded if and only if <i>h(y)</i> is a bounded operator from <i>E</i> onto <i>F</i> for all <i>y</i> in <i>Y</i>. In this case, <i>h</i> is continuous with respect to the strong operator topology.

Let and be a real uniformly smooth Banach space. Let be a nonempty closed convex subset of and be a strictly pseudocontractive mapping in the sense of FE Browder and WV Petryshyn [1]. Let be a bounded sequence in and be real sequences in satisfying some restrictions. Let be the bounded sequence in generated from any given by the Ishikawa iteration method with errors: , , . It is shown in this paper that if is compact or demicompact, then converges strongly to a fixed point of .

We analyze a family of singular Schrdinger operators with local singular interactions supported by a hypersurface n, n 2, being the boundary of a Lipschitz domain, bounded or unbounded, not necessarily connected. At each point of the interaction is characterized by four real parameters, the earlier studied case of - and -interactions being particular cases. We discuss spectral properties of these operators and derive operator inequalities between those referring to the same hypersurface but different couplings and describe their implications for spectral properties.

Our aim is to study fractional order differential inclusions with infinite delays in Banach spaces. We impose the regularity condition on multivalued nonlinearity in terms of measures of noncompactness to get the existence result. Some properties of the solution map are proved.

We demonstrate a one-dimensional magnetic system can exhibit a Cantor-type spectrum using an example of a chain graph with coupling at the vertices exposed to a magnetic field perpendicular to the graph plane and varying along the chain. If the field grows linearly with an irrational slope, measured in terms of the flux through the loops of the chain, we demonstrate the character of the spectrum relating it to the almost Mathieu operator.

We analyze two-dimensional Schrdinger operators with the potential |{xy}{|}^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+ 2)} where |{xy}{|}^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+ 2)} and |{xy}{|}^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+ 2)} which exhibit an abrupt change of spectral properties at a critical value of the coupling constant . We show that in the supercritical case the spectrum covers the whole real axis. In contrast, for below the critical value the spectrum is purely discrete and we establish a LiebThirring-type bound on its moments. In the critical case where the essential spectrum covers the positive halfline while the negative spectrum can only be discrete, we demonstrate numerically the existence of a ground-state eigenvalue.