Comments on completely continuous operators and Fredholm determinants
نویسنده
چکیده
Completely continuous operators Let A be a Banach algebra with |A| % {1}, where A := {a ∈ A : |ab| = |a| |b| for all b ∈ A} is the set of all multiplicative units in A (equivalently, A = {a ∈ A : |a| |a| = 1}). Then for any Banach modules M,N over A, an A-linear map L : M → N is continuous iff supm6=0 |L(m)| |m| < ∞. Let BA(M,N) be the space of such maps. With the norm |L| := supm6=0 |L(m)| |m| , BA(M,N) is a Banach A-module. For a Banach A-module M , let M = BA(M,A) be the continuous dual of M . As in the algebraic case, there is a natural pairing
منابع مشابه
Toeplitz Operators and Toeplitz Algebra with Symbols of Vanishing Oscillation
We study the C∗-algebra generated by Teoplitz operators with symbols of vanishing (mean) oscillation on the Bergman space of the unit ball. We show that the index calculation for Fredholm operators in this C∗-algebra can be easily and completely reduced to the classic case of Toeplitz operators with symbols that are continuous on the closed unit ball. Moreover, in addition to a number of other ...
متن کامل(modified) Fredholm Determinants for Operators with Matrix-valued Semi-separable Integral Kernels Revisited
We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Green’s functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) Fredholm determinants in terms of solutions of closely associate...
متن کاملOn Dirichlet-to-neumann Maps and Some Applications to Modified Fredholm Determinants
We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrödinger operators in L2(Ω; dnx), where Ω ⊂ Rn, n = 2, 3, are open sets with a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. As an application we describe a reduction of a certain ratio of modified Fredholm perturbation determinants associated with operators in L2(Ω; dnx) to modifie...
متن کاملCompletely continuous endomorphisms of p-adic Banach spaces
In Dwork’s memoir [3] concerning the rationality of zeta functions, an essential role is played by the p-adic analytic function det(1− tu), where u is a certain infinite matrix. This analytic function is an entire function, exactly as in the classical Fredholm theory. It was natural to pursue this analogy and extend to u the spectral theory of F. Riesz; this is just what Dwork did ([4], §2). In...
متن کاملar X iv : m at h - ph / 0 11 10 08 v 1 5 N ov 2 00 1 DISCRETE GAP PROBABILITIES AND DISCRETE PAINLEVÉ EQUATIONS
We prove that Fredholm determinants of the form det(1 − Ks), where Ks is the restriction of either the discrete Bessel kernel or the discrete 2F1 kernel to {s, s + 1, . . . }, can be expressed through solutions of discrete Painlevé II and V equations, respectively. These Fredholm determinants can also be viewed as distribution functions of the first part of the random partitions distributed acc...
متن کامل