Let L := −r−2(r∂r)2 − ∂2 z . We consider the equation Lu = f on a bounded polygonal domain with suitable boundary conditions, derived from the three-dimensional axisymmetric Poisson’s equation. We establish the well-posedness, regularity and Fredholm results in weighted Sobolev spaces, for possible singular solutions caused by the singular coefficient of the operator L, as r → 0, and by non-smo...

We show that the Fredholm spectrum of an evolution semigroup {E}t≥0 is equal to its spectrum, and prove that the ranges of the operator Et − I and the generator G of the evolution semigroup are closed simultaneously. The evolution semigroup is acting on spaces of functions with values in a Banach space, and is induced by an evolution family that could be the propagator for a well-posed linear d...

This paper presents direct settings and rigorous solutions of the main Statistical Inference problems. It shows that rigorous solutions require solving multidimensional Fredholm integral equations of the first kind in the situation where not only the right-hand side of the equation is an approximation, but the operator in the equation is also defined approximately. Using Stefanuyk-Vapnik theory...

We give a short proof of the theorem of Brown, Douglas and Fillmore that an essentially normal operator on a Hilbert space is a compact perturbation of a normal operator if and only if the Fredholm index of its translates are zero whenever deened. The proof is a modiication of the short proof of Lin's theorem on almost commuting self-adjoint elements we gave in 7]. Using similar methods we prov...

The main purpose of this article is to give a relationship between Fredholm multivalued linear operators and the demicompact relation; we provide some sufficient conditions on inputs closable block operator matrix ensure generalized demicompactness its closure. Our results generalize many known ones in literature.

We formulate the generic $\tau$-function of Painlev\'e II equation as a Fredholm determinant an integrable (Its-Izergin-Korepin-Slavnov) operator. The depends on isomonodromic time $t$ and two Stokes' parameters, vanishing locus $\tau$-function, called Malgrange divisor is determined by zeros determinant.

A hypoelliptic operator in the Heisenberg calculus on a compact contact manifold is Fredholm operator. Its symbol determines an element K-theory of noncommutative algebra symbols. We construct periodic cyclic cocycle which, when paired with Connes-Chern character principal symbol, calculates index. Our index formula local, i.e. given as local expression terms and connection TM its curvature. pr...

Theorem 1. Let M be a compact manifold, E and F be two (smooth) vector bundles over M , and P : C∞(M ;E) → C∞(M ;F ) be a differential operator (with smooth coefficients) of order n. Assume that P is elliptic, in the sense defined in the next section. Then: 1. The operator P (or rather its relevant extension) is Fredholm W n 2 (M ;E)→ L(M ;F ). Here W n 2 is a Sobolev space (introduced below). ...

Let G be a Jordan domain and K C G be relatively closed with Area(K) = 0. Let A2(G\K) and A2 (G) be the Bergman spaces on G\K, respectively G and define N = A2(G\K) e A2 (G). In this paper we show that with a mild restriction on K, every function in N has an analytic continuation across the analytic arcs of 8G that do not intersect K. This result will be used to discuss the Fredholm theory of t...

Abstract. The problem of evaluating the information associated with Fredholm integral equations of the first kind, when the integral operator is self– adjoint and compact, is considered here. The data function is assumed to be perturbed gently by an additive noise so that it still belongs to the range of the operator. First we estimate upper and lower bounds for the ε–capacity (and then for the...