Essential range

Abstract: The paper is devoted to some only recently uncovered phenomena emerging in the study of singular integral operators (SIO's) with piecewise continuous (PC) coefficients in reflexive rearrangement-invariant spaces over Carleson curves. We deal with several kinds of indices of submultiplicative functions which describe properties of spaces (Boyd and Zippin indices) and curves (spirality indices). We consider some “disintegration condition” which combines properties of spaces and curves, the Boyd and spirality indices. We show that the essential spectrum of SIO associated with the Riemann boundary value problem with PC coefficient arises from the essential range of the coefficient by filling in certain massive connected sets (so-called logarithmic leaves) between the endpoints of jumps. These results combined with the Allan-Douglas local principle and with the two projections theorem enable us to study the Banach algebra $$\mathfrak{A}$$ generated by SIO's with matrix-valued piecewise continuous coefficients. We construct a symbol calculus for this Banach algebra which provides a Fredholm criterion and gives a basis for an index formula for arbitrary SIO's from $$\mathfrak{A}$$ in terms of their symbols.

Abstract: Introduction. This introduction will present a quick survey of our results; the complete definitions necessary to state these results precisely are given in later sections. Let H°° denote the algebra of bounded analytic functions on the open unit disk in the complex plane. The maximal ideal space of H°° is denoted by M. We can think of the open unit disk as a dense subset of M. Carl Sundberg [11] proved that every function in BMO extends to a continuous function from M into the Riemann sphere; he also described several properties of these extensions. Sundberg was working in the context of functions of several real variables. In the next section of this paper we take advantage of the tools offered by analytic function theory to give considerably simpler proofs (in the context of one complex variable) of Sundberg's results about extensions of BMO functions. In the section of this paper on nontangential limits, we prove that a function on the disk that has a continuous extension to a small subset of M must have a nontangential limit at almost every point of the unit circle. We then use this result to produce a class of functions in the little Bloch space that cannot be extended to. be continuous functions from this small subset of M to the Riemann sphere. The section of this paper dealing with cluster sets and essential ranges shows how those sets can be computed from the appropriate continuous extensions. We use these results to give a new proof of Joel Shapiro's theorem [10] that for every function in VMO, the essential range equals the cluster set. The final section of the paper discusses some open questions.

Abstract: The localization techniques of Douglas and Sarason are used to obtain the essential spectrum of the Toeplitz operator T1, for which q is the product of a continuous function and the characteristic function of a measurable subset of the unit circle. Examples are given of Toeplitz operators with onedimensional self-commutator whose essential spectrum IS the unit disk. Using an example of J. E. Brennan, the authors show the existence of a completely nonnormal, subnormal operator whose adjoint has no point spectrum. Introduction. Let Iu denote normalized Lebesgue measure on the unit circle T and let LP(a)=LP, 1