Proceedings of the American Mathematical Society, Mar 28, 2008
We construct a simple reproducing kernel space whose multiplier algebra does not satisfy a "coron... more We construct a simple reproducing kernel space whose multiplier algebra does not satisfy a "corona theorem".
Let H 2 (μ) denote the closure of the polynomials in L 2 (μ), where μ is a positive finite compac... more Let H 2 (μ) denote the closure of the polynomials in L 2 (μ), where μ is a positive finite compactly supported Borel measure carried by the closed unit disc D. For λ e D, define E(λ)=8up{\p(λ)\/\\p\\ μ }, where the suprenum is taken over all polynomials whose L 2 (μ) norm is not zero. If E(λ)<oo we say that μ has a bounded point evaluation at λ, abbreviated b.p.e. at λ. Whenever E(λ)<oo we may fix the value of feH 2 (μ) at λ. We determine the set on which all functions in H 2 (μ) have (fixed) analytic values in terms of the parts of the spectrum of a certain operator.
A strict maximum modulus theorem is proved for certain Banach spaces. As an application it is sho... more A strict maximum modulus theorem is proved for certain Banach spaces. As an application it is shown that ifAX=XA* withA hyponormal andX a member of any Schattenp-class withp=1, thenA*X=XA.
For a countable number of input functions in H ∞ (D n ), we find explicit analytic solutions belo... more For a countable number of input functions in H ∞ (D n ), we find explicit analytic solutions belonging to the Orlicz-type space, exp(L\tfrac12n -&lt;/font &gt; 1 ) \exp (L^{\tfrac{1} {{2n - 1}}} ) . Note that H¥&lt;/font &gt; (Dn ) -&lt;/font &gt; BMO(Dn ) \subsetneqq exp(L\tfrac12n -&lt;/font &gt; 1 ) \subsetneqq Ç&lt;/font &gt;1¥&lt;/font &gt; Hp (Dn ) H^\infty (D^n ) - BMO(D^n ) \subsetneqq \exp (L^{\tfrac{1} {{2n - 1}}} ) \subsetneqq \cap _1^\infty H^p (D^n )
Integral Equations and Operator Theory, Jul 1, 1990
A geometric condition for the existence of bounded evaluations is given. Using this criterion for... more A geometric condition for the existence of bounded evaluations is given. Using this criterion for sets of finite perimeter, it is shown that P2(wdm~) has analytic bounded point evaluations, when w is of bounded variation with compact support. This theorem is then related to previous work on bounded point evaluations. 0. Trent Borel measures. For ~ a positive measure and 1 < l < | we denote by pl(~) the closure in Ll(#) of the analytic polynomials. We say that pl(#) has a bounded point evaluation (b.p.e.) at the point ] in the complex plane, C, if the linear functional defined on polynomials by, ~ p(2) is bounded as a functional on pl(#). A point 2 is an analytic bounded point evaluation (a.b.p.e.
Proceedings of the American Mathematical Society, May 4, 2007
We extend the "matricial corona theorem" of M. Andersson to general algebras of functions which s... more We extend the "matricial corona theorem" of M. Andersson to general algebras of functions which satisfy a corona theorem.
Proceedings of the American Mathematical Society, Mar 1, 1984
It is shown that if g e C^.(C) with 9g nonvanishing on the support of ¡i and if P2(ii) has no bou... more It is shown that if g e C^.(C) with 9g nonvanishing on the support of ¡i and if P2(ii) has no bounded point evaluations, then sp{ P2(¡i) + gf 2(m)}~= L2()i). Similar theorems stating that in the absence of bounded point evaluations P2(fi) is "almost" L2(fi) are derived. As a consequence, to show that P2(¡i) = L2(¡x) in the absence of bounded point evaluations, one need only show that, for example, i¡z-\ e P2(ii) for complex X's. Let ju. denote a finite positive Borel measure with compact support in the complex plane. Let />2(ju) denote the closure in L2(ju) of the polynomials in z. A question of interest is to determine when P2(fi) = L2(ii). If /t. is supported on the boundary of the unit circle, dD, such a characterization has been given by a classical result of Szegö [5]: either P2(n) = L2(n) or else P2(¡i) has a bounded point evaluation. />2(ju) has a bounded point evaluation (or b.p.e.) at w in the complex plane C whenever there exists a constant C with 0 < C < oo and |/?(w)| < C||/?||2ifl for all p e P2(n). For measures y absolutely continuous with respect to area Lebesgue measure m, a result analogous to Szegö's theorem has been discovered by Brennan [1-4] and Hruschev (see [4]), with the mild hypothesis that dy/dm belong to L(Log+ L)2(m). In this note we show that if P2(fi) has no b.p.e.'s, then />2(ju) and L2(n) cannot differ by "too" much. As a consequence, to show that P2(/x) = L2(¡x) in the absence of bounded point evaluations, one need only show that, for example, •Jz-X e P2(n) for complex X's. Denote the support of ju by K. Let g be a continuously differentiate function on C with 9g nonvanishing on K, where 3 denotes the operator l/2(9x + idy). Let {2,}/e/c£°°(j<0-By sp{giP2(n): i&I) we mean the [Zj.^pf ij^I and Pj^P2{li)îoxj=\,2,...,K}. Theorem 1. Suppose P2(¡i) has no b.p.e.'s. Then sp{/*2(m) + gP2{fi)}'= L2(/x). Proof. Let /e L2(n) with f ± [/,2(/x) + gP2(p)]. Then for all X e C and any polynomial p Q=t p^-pWig{z)-g{X))7U)d,(z).
Proceedings of the American Mathematical Society, 1981
We prove that the rational module ^t(X~ßt is always uniformly dense in C(X) if the compact set X ... more We prove that the rational module ^t(X~ßt is always uniformly dense in C(X) if the compact set X has no interior. Let Ibea compact subset of the complex plane C. Let the module ty,(Xy$m be the space ft(Jf) + %(X)z +■■■ + <&(X)zm = {r0(z) + rx(z)z +■■■ +rm(z)zm},
For a collection of reproducing kernels k which includes those for the Hardy space of the polydis... more For a collection of reproducing kernels k which includes those for the Hardy space of the polydisk and ball and for the Bergman space, k is a complete Pick kernel if and only if the multiplier algebra of H 2 (k) has the Douglas property. Consequences for solving the operator equation AX = Y are examined.
Proceedings of the American Mathematical Society, Apr 7, 2006
We show that a usual corona-type theorem on a space of functions automatically extends to a matri... more We show that a usual corona-type theorem on a space of functions automatically extends to a matrix version.
Proceedings of the American Mathematical Society, Mar 3, 1999
Given a subspace S of operators on a Hilbert space, and given two operators X and Y (not necessar... more Given a subspace S of operators on a Hilbert space, and given two operators X and Y (not necessarily in S), when can we be certain that there is an operator A in S such that AX = Y ? If there is one, can we find some bound for its norm? These questions are the subject of a number of papers, some by the present authors, and mostly restricted to the case where S is a reflexive algebra. In this paper, we relate the broader question involving operator subspaces to the question about reflexive algebras, and we examine a new method of forming counterexamples, which simplifies certain constructions and answers an unresolved question. In particular, there is a simple set of conditions that are necessary for the existence of a solution in the reflexive algebra case; we show that-even in the case where the co-rank of X is one-these conditions are not in general sufficient.
Complex Analysis and Operator Theory, Aug 28, 2008
The main purpose of this paper is to extend and refine some work of Agler-McCarthy and Amar conce... more The main purpose of this paper is to extend and refine some work of Agler-McCarthy and Amar concerning the Corona problem for the polydisk and the unit ball in C n .
Bulletin of The London Mathematical Society, Sep 1, 1981
For a smooth function g, the module &{X)+g<%{X) is dense in C{X) if and only if dt(Z) is dense in... more For a smooth function g, the module &{X)+g<%{X) is dense in C{X) if and only if dt(Z) is dense in C(Z), where X is compact and nowhere dense and Z is the subset of X on which fig vanishes. The "localness" of the module 0l(X) + g<%(X) is also investigated. Let X be a compact subset of the complex plane C and let g be a continuous function on X. We denote by @{X, g) the rational module »{X) + a{X)g = {r o (z) + r 1 (z)g(z)}, where each r k denotes a rational function with poles off X. The purpose of this paper is to study the uniform closure of such modules on X. In the case that g{z) = z, the closure of 0t{X,z) in various topologies was considered in [3], [4] and was applied to rational approximation problems in lip a. Further results were obtained in [6], [7]. A question which arose from these investigations concerned the characterization of the uniform closure of ${X, z) in C(X) when X has empty interior. This was settled in [5] by showing that <%{X, z) is uniformly dense in C(X) whenever X has empty interior.
We show that if \(F = (f_{1},\ldots,f_{m})\), where f i is in \(H^{\infty }(D^{n})\) and if \(T_{... more We show that if \(F = (f_{1},\ldots,f_{m})\), where f i is in \(H^{\infty }(D^{n})\) and if \(T_{f_{i}}\) denotes the analytic Toeplitz operator acting on \(\mathcal{H} = H^{p}(D^{n})\) for 1 < p < ∞, then $$\displaystyle{\sigma _{T}((T_{f_{1}},\ldots,T_{f_{m}}),\mathcal{H}) = \overline{\text{ran}\,F(D^{n})}^{\mathbb{C}^{m} }.}$$ Here σ T denotes the Taylor spectrum.
Proceedings of the American Mathematical Society, Feb 1, 1981
A Muntz-Szasz theorem is proved for the continuous functions on the closed unit disc ¿.Asa coroll... more A Muntz-Szasz theorem is proved for the continuous functions on the closed unit disc ¿.Asa corollary it is shown that if gcd(«, ni)-1, the uniformly closed algebra generated by I, z", zm is C(D) (Minsker). D
Proceedings of the American Mathematical Society, Mar 28, 2008
We construct a simple reproducing kernel space whose multiplier algebra does not satisfy a "coron... more We construct a simple reproducing kernel space whose multiplier algebra does not satisfy a "corona theorem".
Let H 2 (μ) denote the closure of the polynomials in L 2 (μ), where μ is a positive finite compac... more Let H 2 (μ) denote the closure of the polynomials in L 2 (μ), where μ is a positive finite compactly supported Borel measure carried by the closed unit disc D. For λ e D, define E(λ)=8up{\p(λ)\/\\p\\ μ }, where the suprenum is taken over all polynomials whose L 2 (μ) norm is not zero. If E(λ)<oo we say that μ has a bounded point evaluation at λ, abbreviated b.p.e. at λ. Whenever E(λ)<oo we may fix the value of feH 2 (μ) at λ. We determine the set on which all functions in H 2 (μ) have (fixed) analytic values in terms of the parts of the spectrum of a certain operator.
A strict maximum modulus theorem is proved for certain Banach spaces. As an application it is sho... more A strict maximum modulus theorem is proved for certain Banach spaces. As an application it is shown that ifAX=XA* withA hyponormal andX a member of any Schattenp-class withp=1, thenA*X=XA.
For a countable number of input functions in H ∞ (D n ), we find explicit analytic solutions belo... more For a countable number of input functions in H ∞ (D n ), we find explicit analytic solutions belonging to the Orlicz-type space, exp(L\tfrac12n -&lt;/font &gt; 1 ) \exp (L^{\tfrac{1} {{2n - 1}}} ) . Note that H¥&lt;/font &gt; (Dn ) -&lt;/font &gt; BMO(Dn ) \subsetneqq exp(L\tfrac12n -&lt;/font &gt; 1 ) \subsetneqq Ç&lt;/font &gt;1¥&lt;/font &gt; Hp (Dn ) H^\infty (D^n ) - BMO(D^n ) \subsetneqq \exp (L^{\tfrac{1} {{2n - 1}}} ) \subsetneqq \cap _1^\infty H^p (D^n )
Integral Equations and Operator Theory, Jul 1, 1990
A geometric condition for the existence of bounded evaluations is given. Using this criterion for... more A geometric condition for the existence of bounded evaluations is given. Using this criterion for sets of finite perimeter, it is shown that P2(wdm~) has analytic bounded point evaluations, when w is of bounded variation with compact support. This theorem is then related to previous work on bounded point evaluations. 0. Trent Borel measures. For ~ a positive measure and 1 < l < | we denote by pl(~) the closure in Ll(#) of the analytic polynomials. We say that pl(#) has a bounded point evaluation (b.p.e.) at the point ] in the complex plane, C, if the linear functional defined on polynomials by, ~ p(2) is bounded as a functional on pl(#). A point 2 is an analytic bounded point evaluation (a.b.p.e.
Proceedings of the American Mathematical Society, May 4, 2007
We extend the "matricial corona theorem" of M. Andersson to general algebras of functions which s... more We extend the "matricial corona theorem" of M. Andersson to general algebras of functions which satisfy a corona theorem.
Proceedings of the American Mathematical Society, Mar 1, 1984
It is shown that if g e C^.(C) with 9g nonvanishing on the support of ¡i and if P2(ii) has no bou... more It is shown that if g e C^.(C) with 9g nonvanishing on the support of ¡i and if P2(ii) has no bounded point evaluations, then sp{ P2(¡i) + gf 2(m)}~= L2()i). Similar theorems stating that in the absence of bounded point evaluations P2(fi) is "almost" L2(fi) are derived. As a consequence, to show that P2(¡i) = L2(¡x) in the absence of bounded point evaluations, one need only show that, for example, i¡z-\ e P2(ii) for complex X's. Let ju. denote a finite positive Borel measure with compact support in the complex plane. Let />2(ju) denote the closure in L2(ju) of the polynomials in z. A question of interest is to determine when P2(fi) = L2(ii). If /t. is supported on the boundary of the unit circle, dD, such a characterization has been given by a classical result of Szegö [5]: either P2(n) = L2(n) or else P2(¡i) has a bounded point evaluation. />2(ju) has a bounded point evaluation (or b.p.e.) at w in the complex plane C whenever there exists a constant C with 0 < C < oo and |/?(w)| < C||/?||2ifl for all p e P2(n). For measures y absolutely continuous with respect to area Lebesgue measure m, a result analogous to Szegö's theorem has been discovered by Brennan [1-4] and Hruschev (see [4]), with the mild hypothesis that dy/dm belong to L(Log+ L)2(m). In this note we show that if P2(fi) has no b.p.e.'s, then />2(ju) and L2(n) cannot differ by "too" much. As a consequence, to show that P2(/x) = L2(¡x) in the absence of bounded point evaluations, one need only show that, for example, •Jz-X e P2(n) for complex X's. Denote the support of ju by K. Let g be a continuously differentiate function on C with 9g nonvanishing on K, where 3 denotes the operator l/2(9x + idy). Let {2,}/e/c£°°(j<0-By sp{giP2(n): i&I) we mean the [Zj.^pf ij^I and Pj^P2{li)îoxj=\,2,...,K}. Theorem 1. Suppose P2(¡i) has no b.p.e.'s. Then sp{/*2(m) + gP2{fi)}'= L2(/x). Proof. Let /e L2(n) with f ± [/,2(/x) + gP2(p)]. Then for all X e C and any polynomial p Q=t p^-pWig{z)-g{X))7U)d,(z).
Proceedings of the American Mathematical Society, 1981
We prove that the rational module ^t(X~ßt is always uniformly dense in C(X) if the compact set X ... more We prove that the rational module ^t(X~ßt is always uniformly dense in C(X) if the compact set X has no interior. Let Ibea compact subset of the complex plane C. Let the module ty,(Xy$m be the space ft(Jf) + %(X)z +■■■ + <&(X)zm = {r0(z) + rx(z)z +■■■ +rm(z)zm},
For a collection of reproducing kernels k which includes those for the Hardy space of the polydis... more For a collection of reproducing kernels k which includes those for the Hardy space of the polydisk and ball and for the Bergman space, k is a complete Pick kernel if and only if the multiplier algebra of H 2 (k) has the Douglas property. Consequences for solving the operator equation AX = Y are examined.
Proceedings of the American Mathematical Society, Apr 7, 2006
We show that a usual corona-type theorem on a space of functions automatically extends to a matri... more We show that a usual corona-type theorem on a space of functions automatically extends to a matrix version.
Proceedings of the American Mathematical Society, Mar 3, 1999
Given a subspace S of operators on a Hilbert space, and given two operators X and Y (not necessar... more Given a subspace S of operators on a Hilbert space, and given two operators X and Y (not necessarily in S), when can we be certain that there is an operator A in S such that AX = Y ? If there is one, can we find some bound for its norm? These questions are the subject of a number of papers, some by the present authors, and mostly restricted to the case where S is a reflexive algebra. In this paper, we relate the broader question involving operator subspaces to the question about reflexive algebras, and we examine a new method of forming counterexamples, which simplifies certain constructions and answers an unresolved question. In particular, there is a simple set of conditions that are necessary for the existence of a solution in the reflexive algebra case; we show that-even in the case where the co-rank of X is one-these conditions are not in general sufficient.
Complex Analysis and Operator Theory, Aug 28, 2008
The main purpose of this paper is to extend and refine some work of Agler-McCarthy and Amar conce... more The main purpose of this paper is to extend and refine some work of Agler-McCarthy and Amar concerning the Corona problem for the polydisk and the unit ball in C n .
Bulletin of The London Mathematical Society, Sep 1, 1981
For a smooth function g, the module &{X)+g<%{X) is dense in C{X) if and only if dt(Z) is dense in... more For a smooth function g, the module &{X)+g<%{X) is dense in C{X) if and only if dt(Z) is dense in C(Z), where X is compact and nowhere dense and Z is the subset of X on which fig vanishes. The "localness" of the module 0l(X) + g<%(X) is also investigated. Let X be a compact subset of the complex plane C and let g be a continuous function on X. We denote by @{X, g) the rational module »{X) + a{X)g = {r o (z) + r 1 (z)g(z)}, where each r k denotes a rational function with poles off X. The purpose of this paper is to study the uniform closure of such modules on X. In the case that g{z) = z, the closure of 0t{X,z) in various topologies was considered in [3], [4] and was applied to rational approximation problems in lip a. Further results were obtained in [6], [7]. A question which arose from these investigations concerned the characterization of the uniform closure of ${X, z) in C(X) when X has empty interior. This was settled in [5] by showing that <%{X, z) is uniformly dense in C(X) whenever X has empty interior.
We show that if \(F = (f_{1},\ldots,f_{m})\), where f i is in \(H^{\infty }(D^{n})\) and if \(T_{... more We show that if \(F = (f_{1},\ldots,f_{m})\), where f i is in \(H^{\infty }(D^{n})\) and if \(T_{f_{i}}\) denotes the analytic Toeplitz operator acting on \(\mathcal{H} = H^{p}(D^{n})\) for 1 < p < ∞, then $$\displaystyle{\sigma _{T}((T_{f_{1}},\ldots,T_{f_{m}}),\mathcal{H}) = \overline{\text{ran}\,F(D^{n})}^{\mathbb{C}^{m} }.}$$ Here σ T denotes the Taylor spectrum.
Proceedings of the American Mathematical Society, Feb 1, 1981
A Muntz-Szasz theorem is proved for the continuous functions on the closed unit disc ¿.Asa coroll... more A Muntz-Szasz theorem is proved for the continuous functions on the closed unit disc ¿.Asa corollary it is shown that if gcd(«, ni)-1, the uniformly closed algebra generated by I, z", zm is C(D) (Minsker). D
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