Let C denote the ̄eld of complex numbers and n the set of nth roots of unity. For t = 0; : : : ; n ¡ 1, de ̄ne the ideal =(n; t +1) 1⁄2 C [x0; : : : ; xt] consisting of those polynomials in t + 1 variables that vanish on distinct nth roots of unity; that is, f 2=(n; t+ 1) if and only if f (!0 ; : : : ; !t) = 0 for all (!0 ; : : : ; !t) 2 t+1 n satisfying !i 6= !j, for 0 · i < j · t. In this paper...

An iterative algorithm for computing the principal nth root of a positive deenite matrix is presented. The algorithm is based on the Gauss-Legendre approximation of a deenite integral. We p resent a parallelization in which we use as many processors as the order of the approximation. An analysis of the error introduced at each step of the iteration indicates that the algorithm converges more ra...

Let sn(T ) denote the nth approximation, isomorphism, Gelfand, Kolmogorov or Bernstein number of the Hardy-type integral operator T given by

We characterize hyponormal “rational” Toeplitz pairs which are pairs of Toeplitz operators whose symbols are rational functions in L∞. The main result of this article is as follows. If T = (Tφ, Tψ) is a hyponormal rational Toeplitz pair then φ− βψ ∈ H for some constant β; in other words, their co-analytic parts necessarily coincide up to a constant multiple. As a corollary we get a complete cha...

Abstract We show that if a densely defined closable operator A is such the resolvent set of 2 nonempty, then necessarily closed. This result extended to case polynomial . also generalize recent by Sebestyén–Tarcsay concerning converse J. von Neumann. Other interesting consequences are given. One them proof T quasinormal (unbounded) normal for some , normal. Hence closed subnormal itself hyponor...

Let B(H) be the set of all bounded endomorphisms (operators) on the complex Hubert space H. T € B(H) is paranormal if \\(T — z/) || = l/d(z, cr(T)) for all z £ cr(T) where d(z, cr(T)) is the distance from z to cr(T), the spectrum of T. If J is the set of all paranormal operators on H, then J contains the normal operators, JÍ, and the hyponormal operators; and J is contained in JL, the set of al...

Index theory has had profound impact on many branches of mathematics. In this note we discuss the context for a new kind of index theorem. We begin, however, with some operator-theoretic results. In [11] Berger and Shaw established that finitely cyclic hyponormal operators have trace-class self-commutators. In [9], [31] Berger and Voiculescu extended this result to operators whose self-commutat...