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This paper is concerned with Toeplitz matrices generated by symbols of the form a(t) = Φ r=1 R | t−t r | 2α r (− t ) t r β r b(t) (| t |=1) where t 1 ,…, t R are pairwise distinct points on the unit circle, b is sufficiently smooth, b(t) ≠ 0 (¦t¦ = 1) , and ind b = 0, (− t ) t r β r is defined as exp {iβ r arg ( − t t r )} with ¦arg ( − t t r )¦ , and α r , β r are complex numbers satisfying ¦ Re α r ¦ 1 2 , ¦ Re β r ¦ 1 2 . Weighted Hardy spaces L + 2 ( ϱ 1 ) and L + 2 ( ϱ 2 ) are defined such that the Toeplitz operator T ( a ): L + 2 ( ϱ 1 ) → L + 2 ( ϱ 2 ) is bounded and invertible and that the finite section method is applicable to T ( a ) considered as acting from L + 2 ( ϱ 1 ) onto L + 2 ( ϱ 2 ). These results are then applied to prove a conjecture of Fisher and Hartwig ( Adv. in Chem. Phys. 15 (1968), 333–353) which asserts that the determinants of the n × n sections of T ( a ) are asymptotically equal to G(b) n n q E (a) (n → ∞) , where q = ∑ ( α r 2 − β r 2 ) and G ( b ), E (a) are certain nonzero constants. Under quite general conditions on the smoothness of b, these constants are completely identified.

Local principles associate with every element of a Banach algebra a family of local objects in terms of which the properties of the original element can be studied. In this paper some general relations between three such principles, usually affiliated with the names of Simonenko, Allan/Douglas, and Gohberg/Krupnik, are discussed. Special attention is paid to the question on how the norm of an element can be expressed in terms of the norms of the local objects associated with it. The general theory is illustrated by some concrete results on singular integral operators and Toeplitz operators.

Criteria for Drazin and Moore–Penrose invertibility of operators in the von Neumann algebra generated by two orthogonal projections are established and explicit representations for the corresponding inverses are given. The results are illustrated by several examples that have recently been considered in the literature.

The spectral value set spεB,CA of a bounded linear operator A on l2 is the union of the spectra of all operators of the form A+BKC, where ∥K∥

This paper is a survey of the basics of the theory of two projections. It contains in particular the theorem by Halmos on two orthogonal projections and Roch, Silbermann, Gohberg, and Krupnik’s theorem on two idempotents in Banach algebras. These two theorems, which deliver the desired results usually very quickly and comfortably, are missing or wrongly cited in many recent publications on the topic, The paper is intended as a gentle guide to the field. The basic theorems are precisely stated, some of them are accompanied by full proofs, others not, but precise references are given in each case, and many examples illustrate how to work with the theorems.

In an earlier paper we conjectured an inequality for the Frobenius norm of the commutator of two matrices. This conjecture was recently proved by Seak-Weng Vong and Xiao-Qing Jin and independently also by Zhiqin Lu. We here give a completely different proof of this inequality, prove some related results, and embark on the corresponding question for unitarily invariant norms.

The subject of this paper is polynomial collocation for approximately solving Toeplitz integral equations on the Bergman space of the complex unit disk. It turns out that the convergence of such methods depends on the choice of the system of collocation points. We establish a result for the case where the collocation points are evenly distributed on massive subsets of the unit disk.