The parameter imbedding method for the Fredholm equation converts it into an initial value problem in its parameter λ. We establish the method for general operator equations of the form [I + ˆ f (λ)]ψ = φ. It is particularly useful for studying spontaneous symmetry breaking problems, such as contained in the nonlinear Schwinger-Dyson equation.

We give the Jordan form and the Singular Value Decomposition for an integral operator N with a non-symmetric kernel N(y, z). This is used to give solutions of Fredholm equations for non-symmetric kernels, and to determine the behaviour of N n and (NN ∗)n for large n.

Correlation functions of the XXZ spin chain in the critical regime is studied at zerotemperature. They are exactly represented in the Fredholm determinant form and are related with an operator-valued Riemann–Hilbert problem. Analyzing this problem we prove that a two-point correlation function consisting of sufficiently separated spin operators is expressed by power-functions of the distance be...

The famous Hirzebruch signature theorem asserts that the signature of a closed oriented manifold is equal to the integral of the so called L-genus. An immediate corollary of this is the homotopy invariance of < L(M); [M ] >. The L-genus is a characteristic class of tangent bundles, so the above remark is a non-trivial fact. The problem of higher signatures is a generalization of the above consi...

Schwinger terms of current algebra can be identified with nontrivial cyclic cocycles of a Fredholm module. We discuss its temperature dependence. Similar anomalies may occur also in spin systems. In simple examples already an operator–valued cocycle shows up. Lectures given at the XXX–th Karpacz Winter School in Theoretical Physics, Poland, 1994.

For a wide class of weights we find the approximative point spectrum and the essential spectrum of the pointwise multiplication operator Mφ, Mφ(f)=φf , on the weighted Banach spaces of analytic functions on the disc with the sup-norm. Therefore we characterize when Mφ is Fredholm or is an isomorphism into. We study also cyclic phenomena of the adjoint map M ′ φ.

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 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 consider the 1D massless Dirac operator on the real line with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following properties of the resonances: 1) asymptotics of counting function, 2) estimates on the resonances and the forbidden domain, 3) the trace formula in terms of re...