The quantum nonrelativistic two-component Bose and Fermi gases with the infinitely strong point-like coupling between particles in one space dimension are considered. Time and temperature dependent correlation functions are represented in the thermodynamic limit as Fredholm determinants of integrable linear integral operators. E-mail: [email protected] E-mail: [email protected]

The authors show that a wide class of Fredholm determinants arising in the representation theory of " big " groups such as the infinite–dimensional unitary group, solve Painlevé equations. Their methods are based on the theory of integrable operators and the theory of Riemann–Hilbert problems.

We consider Toeplitz operators with symbols that are almost periodic matrix functions of several variables. It is shown that under certain conditions on the group generated by the Fourier support of the symbol, a Toeplitz operator is Fredholm if and only if it is invertible.

We apply an analogue of the Zakharov-Shabat dressing method to obtain infinite matrix solutions to the Toda lattice hierarchy. Using an operator transformation we convert some of these into solutions in terms of integral operators and Fredholm determinants. Others are converted into a class of operator solutions to the l-periodic Toda hierarchy.

We consider Fredholm determinants of matrix Hankel operators associated to versions the n-th Airy functions. Using theory integrable operators, we relate them a fully noncommutative Painlevé II hierarchy, defined through matrix-valued version Lenard operators. In particular, Riemann-Hilbert techniques used study these allows find Lax pair for each member hierarchy. Finally, coefficients matrice...

We discuss some of the recent progress in the field of Toeplitz operators acting on Bergman spaces of the unit disk, formulate some new results, and describe a list of open problems—concerning boundedness, compactness and Fredholm properties—which was presented at the conference “Recent Advances in Function Related Operator Theory” in Puerto Rico in March 2010.

We suggest the construction of an oriented coincidence index for nonlinear Fredholm operators of zero index and approximable multivalued maps of compact and condensing type. We describe the main properties of this characteristic, including applications to coincidence points. An example arising in the study of a mixed system, consisting of a first-order implicit differential equation and a diffe...

In this paper we are concerned with the spectral analysis for classes of finte rank perturbations diagonal operators in form A = D+F where D is a operator and F u’1 ⊗v’1+...+u’m ⊗vm an finite non-archimedean Banach space countable type. We compute spectrum using theory Fredholm non archimedean setting concept essential linear operators.