Polydisc Algebra Research Articles

Spectrum of weighted composition operators part X: the spectrum and essential spectra of weighted automorphisms of the polydisc algebra

Spectrum of weighted composition operators part X: the spectrum and essential spectra of weighted automorphisms of the polydisc algebra

Necessary conditions on the support of RP-measures

By associating RP-measures with annihilators of A0→ – a subspace of the polydisc algebra – we provide necessary conditions for when a compact subset of Tn can contain the support of a non-zero RP-measure. Among other things we show that the support of a positive RP-measure cannot be contained in reflections of inverse images of half-planes by functions in A0→, sets of linear measure zero, and when n=2, graphs of strictly increasing functions. Furthermore, we prove that failure of a compact set S to support any positive annihilator of A0→ is equivalent to uniform approximability of all continuous functions on S by functions in A0→. For n=2 this gives a necessary and sufficient condition for the support of positive RP-measures.

Norm Attaining Elements of the Ball Algebra H∞(BN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^\\infty (B_N)$$\\end{document}

Let BN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_N$$\\end{document} be the Euclidean ball of CN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}^N$$\\end{document}. The space H∞(BN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^\\infty (B_N)$$\\end{document} of bounded holomorphic functions on BN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_N$$\\end{document} is known to have a predual, denoted by G∞(BN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G^\\infty (B_N)$$\\end{document}. We study the functions in H∞(BN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^\\infty (B_N)$$\\end{document} that attain their norm as elements of the dual of G∞(BN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G^\\infty (B_N)$$\\end{document}. We also examine similar questions for the polydisc algebra H∞(DN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^\\infty ({\\mathbb {D}}^N)$$\\end{document} and for the space of Dirichlet series D∞(C+).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal {D}}^\\infty ({\\mathbb {C}}_+).$$\\end{document}

Vector-valued holomorphic functions in several variables

In the present paper we give some explicit proofs for folklore theorems on holomorphic functions in several variables with values in a locally complete locally convex Hausdorff space $E$ over $\mathbb{C}$. Most of the literature on vector-valued holomorphic functions is either devoted to the case of one variable or to infinitely many variables whereas the case of (finitely many) several variables is only touched or is subject to stronger restrictions on the completeness of $E$ like sequential completeness. The main tool we use is Cauchy's integral formula for derivatives for an $E$-valued holomorphic function in several variables which we derive via Pettis-integration. This allows us to generalise the known integral formula, where usually a Riemann-integral is used, from sequentially complete $E$ to locally complete $E$. Among the classical theorems for holomorphic functions in several variables with values in a locally complete space $E$ we prove are the identity theorem, Liouville's theorem, Riemann's removable singularities theorem and the density of the polynomials in the $E$-valued polydisc algebra.

Fredholmness of Toeplitz operators on generalized Hardy spaces over the polydisc

We study the structure of the C*-algebra C*(A) generated by a system A of unilateral weighted shifts when the operator algebra generated by A is the polydisc algebra. With help of the obtained results, we present a necessary and sufficient condition for the Toeplitz operators with continuous symbols to be Fredholm.

A short proof of Cartan’s Nullstellensatz for entire functions in $${\mathbb{C}^n}$$ C n

Using the fact that the maximal ideals in the polydisk algebra are given by the kernels of point evaluations, we derive a simple formula that gives a solution to the Bézout equation in the space of all entire functions of several complex variables. Thus a short and easy analytic proof of Cartan’s Nullstellensatz is obtained.

Topological genericity of nowhere differentiable functions in the disc and polydisc algebras

In this paper we examine functions in the disc algebra A(D) and the polydisc algebra A(DI), where I is a finite or countably infinite set. We prove that, generically, for every f∈A(D) the continuous periodic functions u=Ref|T and u˜=Imf|T are nowhere differentiable on the unit circle T. Afterwards, we generalize this result by proving that, generically, for every f∈A(DI), where I is as above, the continuous periodic functions u=Ref|TI and u˜=Imf|TI have no directional derivatives at any point of TI and every direction v∈RI with ‖v‖∞=1. Finally, we describe how our proofs can be modified to give similar results for nowhere Hölder functions in these algebras.

Noncoherence of some rings of holomorphic functions in several variables as an easy consequence of the one-variable case

Using the facts that the disk algebra and the Wiener algebra are not coherent, we prove that the polydisc algebra, the ball algebra, and the Wiener algebra of the polydisc are not coherent.

Contractive Hilbert modules and their dilations

In this note, we show that a quasi-free Hilbert module R defined over the polydisk algebra with kernel function k(z,w) admits a unique minimal dilation (actually an isometric co-extension) to the Hardy module over the polydisk if and only if S −1(z, w)k(z, w) is a positive kernel function, where S(z,w) is the Szegö kernel for the polydisk. Moreover, we establish the equivalence of such a factorization of the kernel function and a positivity condition, defined using the hereditary functional calculus, which was introduced earlier by Athavale [8] and Ambrozie, Englis and Müller [2]. An explicit realization of the dilation space is given along with the isometric embedding of the module R in it. The proof works for a wider class of Hilbert modules in which the Hardy module is replaced by more general quasi-free Hilbert modules such as the classical spaces on the polydisk or the unit ball in ℂm. Some consequences of this more general result are then explored in the case of several natural function algebras.

Duality, Tangential Interpolation, and Töplitz Corona Problems

In this paper we extend a method of Arveson and McCullough to prove a tangential interpolation theorem for subalgebras of $H^\infty$. This tangential interpolation result implies a Toelitz corona theorem. In particular, it is shown that the set of matrix positivity conditions is indexed by cyclic subspaces, which is analogous to the results obtained for the ball and the polydisk algebra by Trent-Wick and Douglas-Sarkar.

Bass and Topological Stable Ranks of Complex and Real Algebras of Measures, Functions and Sequences

We compute the Bass stable rank and the topological stable rank of several convolution Banach algebras of complex measures on (-∞,∞) or on [0,∞) consisting of a discrete measure and/or of an absolutely continuous measure. We also compute the stable ranks of the convolution algebras , , l1(S) and , where S is an arbitrary subgroup of , of the almost periodic algebra AP and of , etc. We answer affirmatively the question posed by Mortini (Studia Mathematica 103(3):275–281, 1992). For the above algebras, the polydisc algebra , the algebra of continuous functions, and others, we also study their subsets (real Banach algebras) of real-valued measures, real-valued sequences or real-symmetric functions, and of corresponding exponentially stable algebras (for example, the Callier–Desoer algebra of causal exponentially decaying measures and L1 functions), and we compute their stable ranks. Finally, we show that in some of these real algebras a variant of the parity interlacing property is equivalent to reducibility of a unimodular (or coprime) pair. Also corona theorems are presented and the existence of coprime fractions is studied; in particular, we list which of these algebras are Bezout domains.

Bourgain algebras and tightness in some algebras of analytic functions

We prove that the Bourgain algebra of the polydisk algebra A ( Δ n ) is A ( Δ n ) itself and disprove the tightness of some algebras of analytic functions; in particular that of H ∞ ( B E ) .

Schur coefficients in several variables

In the paper we study weakly continuous Schur-class-valued maps and their associated Schur coefficient families, that we call functional Schur coefficients. A case of special interest is the family of the “slices” through the polytorus of an n-variable function in the unit ball of H ∞ ( D n ) , which is shown to be a weakly continuous map from the polytorus into the Schur class. The continuity properties of its functional Schur coefficients are used to characterize the rational inner functions in the polydisk algebra. As a consequence we obtain extensions in several variables of the Schur–Cohn test on zeroes of polynomials. This provides in particular a necessary and sufficient condition of stability for multi-dimensional AR filters.

Holomorphic Jackson's theorems in polydiscs

The purpose of this article is to establish Jackson-type inequality in the polydiscs U N of C N for holomorphic spaces X , such as Bergman-type spaces, Hardy spaces, polydisc algebra and Lipschitz spaces. Namely, E k ⇒ ( f , X ) ≲ ω 1 / k → , f , X , where E k ⇒ ( f , X ) is the deviation of the best approximation of f ∈ X by polynomials of degree at most k j about the jth variable z j with respect to the X -metric and ω ( 1 / k → , f , X ) is the corresponding modulus of continuity.

EXTENSIONS OF HARDY MODULES OVER THE POLYDISK ALGEBRA

In this paper, the Ext groups of Hardy modules over the polydisk algebra A(Dn) are calculated. Of particular importance is that the calculating of Ext-groups is closely related to harmonic analysis of polydisks. Finally, the authors point out that Ext-groups reveal rigidity of Hardy submodules over A(Dn) for n > 1.

Closed ideals in the bidisc algebra

Let D be the open unit disc in C, T = 0 D the unit circle, and let D " = D X ... • be the n-dimensional unit polydisc. Let H ~ (D") be the algebra of bounded analytic functions on D", endowed with the uniform norm on D". The polydisc algebra is the space A(D")=C(D")nH~'(D"), also given the uniform norm on D". The spaces A(D) and A(D 2) are known as the disc and bidisc algebras, respectively. Let us introduce a weak-star topology on H = (D"). The space L ~* (T") is the dual space of LI(T"), so it has a weak-star topology. One can think of H*~(D ") as a subspace of L = (T") via radial limits, and as such it is weak-star closed. We define the weak-star topology on H = ( D ") by saying that a set U c H ~ ( D ") is open i f there is a weak-star open set V c L ~ ( T ") with U= VnH~ (D"). For a collection ~" o f functions in A(D"), associate the zero set

On the rigidity of Hardy submodules

In contrast with the one-variable case, there is a large number of distinct submodules of the Hardy module over the polydisk algebra in the multi-variable use. We show that under hypotheses on the zero sets, two submodules which are equivalent in any reasonable sense must be equal. This is the rigidity referred to in the title.

Two observations regarding first-quadrant causal BIBO-stable digital filters

An n-dimensional digital filter is said to be normal if it is linear shift-invariant, first-quadrant causal, and bounded-input-bounded output (BIBO) stable. The transfer function of such a filter is holomorphic in the open polydisc U/sup n/ and continuous in its closure, making it a member of W. Rudin's polydisc algebra A(U/sup n/) (see W. Rudin) (Function Theory in Polydiscs, New York: W.A. Benjamin, 1969). It has been shown by Rudin that, nevertheless, every all-pass function in A(U/sup n/) is rational, free of nonessential singularities of the second kind, and of finite norm. This result appears to have been completely overlooked in the engineering literature. The material covered here is mainly tutorial, comprising a leisurely expanded account of the key ideas behind the Fejer and Rudin constructions. >

Multidimensional digital signal processing: problems, progress, and future scopes

Some of the key mathematical limitations encountered in multidimensional digital signal processing research, the useful general settings in which various topical investigations are facilitated, and directions for future research are summarized. The discussion covers multidimensional sampling, multidimensional signal processing over polydisc algebra, nonlinear multidimensional filtering and prediction, and nonlinear nonstationary multidimensional modelling approaches.

A uniqueness characterization in terms of signed magnitude for functions in the polydisc algebra A(Un): errata

Get PDF Email Share Share with Facebook Tweet This Post on reddit Share with LinkedIn Add to CiteULike Add to Mendeley Add to BibSonomy Get Citation Copy Citation Text Dante C. Youla and S. Unnikrishna Pillai, "A uniqueness characterization in terms of signed magnitude for functions in the polydisc algebra A(Un): errata," J. Opt. Soc. Am. A 6, 1643-1643 (1989) Export Citation BibTex Endnote (RIS) HTML Plain Text Citation alert Save article