In the present paper, we establish first relation between perturbation of upper Fredholm and strictly singular, then lower semi-Fredholm cosingular linear relations. Most importantly in Theorem 3.4, show that P(F?(X,Y)) coincides with SC(X,Y).We bring to light, relationship essential spectra a multivalued operator its selection

In this paper, we show that an unbounded weakly S0-demicompact linear operator T, introduced in [18], acting on a Banach space, can be characterized by some measures of weak noncompactness. Moreover, our results are illustrated to discuss the relationship with Fredholm and upper semi-Fredholm operators as well stability essential spectrum T.

We show that the eigenvalues of the first order partial differential equation derived by quasi-classical approximation of the Schrödinger equation can be computed from the trace of a classical operator. The derived trace formula is different from the Gutzwiller trace formula. The Fredholm determinant of the new operator is an entire function of the complex energy plane in contrast to the semi-c...

Let (U(t))t≥0 be a C0-semigroup of bounded linear operators on a Banach space X. In this paper, we establish that if, for some t0 > 0, U(t0) is a Fredholm (resp., semiFredholm) operator, then (U(t))t≥0 is a Fredholm (resp., semi-Fredholm) semigroup. Moreover, we give a necessary and sufficient condition guaranteeing that (U(t))t≥0 can be embedded in a C0-group on X. Also we study semigroups whi...

Recall some notations and terminology from [4]. For closed subspaces M, L of a Banach space X we write M e ⊂L (M is essentially contained in L ) if there exists a finite-dimensional subspace F ⊂ X such that M ⊂ L + F . Equivalently, dim M/(M ∩L) = dim(M + L)/L < ∞. Similarly we write M e =L if M e ⊂L and L e ⊂M . For a (bounded linear) operator T ∈ L(X) write R∞(T ) = ∞n=0 R(T) and N∞(T ) = ⋃∞ ...

1.1 (Fredholm operators). Let X, Y be real Banach spaces and denote their dual spaces by X, Y . A bounded linear operator D : X → Y is called Fredholm if it has a closed image and if its kernel and cokernel (the quotient space Y/imD) are finite dimensional. Equivalently, there exists a bounded linear operator T : Y → X such that the operators TD− idX and DT − idY are compact. The Fredholm index...

Some basic facts about Fredholm indices are briefly reviewed, often used in connection with Toeplitz and pseudodifferential operators, and which may be relevant for operators associated to fractals. Let H be a complex infinite-dimensional separable Hilbert space, which is of course unique up to isomorphism. Let B(H) be the Banach algebra of bounded linear operators on H, and let C(H) be the clo...

For a continuous nonvanishing complex-valued function g on the real line, several notions of a mean winding number are introduced. We give necessary conditions for a Toeplitz operator with matrix-valued symbol G to be semi-Fredholm in terms of mean winding numbers of det G. The matrix function G is assumed to be continuous on the real line, and no other apriori assumptions on it are made.

For a continuous nonvanishing complex-valued function g on the real line, several notions of a mean winding number are introduced. We give necessary conditions for a Toeplitz operator with matrixvalued symbol G to be semi-Fredholm in terms of mean winding numbers of detG. The matrix function G is assumed to be continuous on the real line, and no other apriori assumptions on it are made. AMS Sub...

In this paper we propose a framework for supervised and semi-supervised learning based on reformulating the learning problem as a regularized Fredholm integral equation. Our approach fits naturally into the kernel framework and can be interpreted as constructing new data-dependent kernels, which we call Fredholm kernels. We proceed to discuss the “noise assumption” for semi-supervised learning ...