Oleksiy Karlovych

We prove asymptotic formulas for determinants and traces of finite block Toeplitz matrices with symbols belonging to Wiener algebras with weights satisfying natural submultiplicativity, monotonicity, and regularity conditions. The remainders in these formulas depend on the weights and go rapidly to zero for very smooth symbols. These formulas refine or extend some previous results by Szegö, Widom, Bottcher, and Silbermann.

A classical theorem of Coifman, Rochberg, and Weiss on commutators of singular integrals is extended to the case of generalized Lp spaces with variable exponent.

We prove Fredholm criteria for singular integral operators of the form
\[
P_++M_bP_-+M_uUP_-,
\]
where \(P_\pm\) are the Riesz projections, \(U\) is the flip operator, and \(M_b,M_u\) are operators of multiplication by functions \(b,u\), respectively, on a reflexive rearrangement-invariant space with nontrivial Boyd indices over the unit circle. We assume a priori that \(M_b\) is bounded, but \(M_u\) may be unbounded. The function \(u\) belongs to a class of, in general, unbounded functions that relates to the Douglas algebra \(H^\infty+C\).

Harold Widom proved in 1966 that the spectrum of a Toeplitz operator \(T(a)\) acting on the Hardy space \(H^p(\mathbb{T})\) over the unit circle \(\mathbb{T}\) is a connected subset of the complex plane for every bounded measurable symbol \(a\) and \(1 < p < \infty\). In 1972, Ronald Douglas established the connectedness of the essential spectrum of \(T(a)\) on \(H^2(\mathbb{T})\). We show that, as was suspected, these results remain valid in the setting of Hardy spaces \(H^p(\Gamma,w)\), \( 1 < p < \infty \), with general Muckenhoupt weights \(w\) over arbitrary Carleson curves \(\Gamma\).

The paper is devoted to the compactness of the commutators \(aS_\Gamma - S_\Gamma aI\) and \(W_\alpha S_\Gamma - S_\Gamma W_\alpha\), where \(S_\Gamma\) is the Cauchy singular integral operator, \(a\) is a bounded measurable function, \(W_\alpha\) is the shift operator given by \(W_\alpha f = f\circ\alpha\), and \(\alpha\) is a bi-Lipschitz homeomorphism (shift). The cases of the unit circle and the unit interval are considered. We prove that these commutators are compact on rearrangement-invariant spaces with nontrivial Boyd indices if and only if the function a or, respectively, the derivative of the shift a has vanishing mean oscillation.

We prove the inverse closedness of the Banach algebra \(\mathfrak{A}_p\) of functional operators with non-Carleman shifts, which have only two fixed points, in the Banach algebra of all bounded linear operators on \(L^p\). We suppose that \(1 \le p \le \infty\) and the generators of the algebra \(\mathfrak{A}_p\) have essentially bounded data. An invertibility criterion for functional operators in \(\mathfrak{A}_p\) is obtained in terms of
the invertibility of a family of discrete operators on \(l^p\). An effective invertibility criterion is established for binomial difference operators with \(l^\infty\) coefficients on the spaces \(l^p\). Using the reduction to binomial difference operators, we give effective criteria of invertibility for binomial functional operators on the spaces \(L^p\).

Suppose \(\alpha\) is an orientation preserving diffeomorphism (shift) of \(\mathbb{R}_+=(0,\infty)\) onto itself with the only fixed points \(0\) and \(\infty\). We establish sufficient conditions for the Fredholmness of the singular integral operator with shift \[ (aI-bW_\alpha)P_++(cI-dW_\alpha)P_- \] acting on \(L^p(\mathbb{R}_+)\) with \( 1 < p < \infty \), where \(P_\pm=(I\pm S)/2\), \(S\) is the Cauchy singular integral operator, and \(W_\alpha f=f\circ\alpha\) is the shift operator, under the assumptions that the coefficients \(a,b,c,d\) and the derivative \(\alpha'\) of the shift are bounded and continuous on \(\mathbb{R}_+\) and may admit discontinuities of slowly oscillating type at \(0\) and \(\infty\).

Let \(\gamma\) be a simple open smooth curve and \(\alpha\) be an orientation-preserving diffeomorphism of \(\gamma\) onto itself which has only two fixed points. Criteria for one-sided invertibility of the functional operator
\[
A=aI-bW,
\]
where \(a\) and \(b\) are continuous functions, \(I\) is the identity operator, \(W\) is the shift operator: \((Wf)(t)=f[\alpha(t)]\), in a reflexive rearrangement-invariant space of fundamental type \(X(\gamma)\) with nontrivial Boyd indices, are obtained.