Holomorphic Kernels and Commuting Operators

Athavale, Ameer

doi:10.2307/2000706

Cited by 22 publications

(31 citation statements)

References 7 publications

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“…The proof has implicitly appeared in the works of Ito [16], Yoshino [31], Lubin [21] and Athavale [3], all dealing with subnormality criteria for commuting tuples of bounded linear operators. Without aiming at completeness, here is the main idea.…”

Section: Example B) In Two Variables We Definementioning

confidence: 99%

Polynomial Optimization on Odd-Dimensional Spheres

D’Angelo

Putinar

2008

Emerging Applications of Algebraic Geometry

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Abstract. The sphere S 2d−1 naturally embeds into the complex affine spaceWe show how the complex variables in C d simplify the known Striktpositivstellensätze, when the supports are resticted to semi-algebraic subsets of odd dimensional spheres. Mathematics Subject Classification (2000). Primary 14P10; Secondary 32A70.Keywords. Positive polynomial, Hermitian square, unit sphere, plurisubharmonic function. PreliminariesLet C d denote complex Euclidean space with Euclidean norm given by |z| 2 = d j=1 |z j | 2 . The unit, odd dimensional sphereis a particularly important example of a Cauchy-Riemann (usually abbreviated CR) manifold. This note will show how one can study problems of polynomial optimization over semi-algebraic subsets of S 2d−1 by using the induced CauchyRiemann structure. Our results can be regarded as multivariate analogues of classical phenomena about positive trigonometric polynomials, known for a long time in dimension one (d = 1). They are also related to results concerning proper holomorphic mappings between balls in different dimensional complex Euclidean spaces and the geometry of holomorphic vector bundles.

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Section: Example B) In Two Variables We Definementioning

confidence: 99%

Polynomial Optimization on Odd-Dimensional Spheres

D’Angelo

Putinar

2008

Emerging Applications of Algebraic Geometry

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“…It was highlighted in [At1] that (G) is equivalent to requiring n → T n h 2 to be completely monotone for every h in H.…”

Section: Non-negative Reals and A Positive Radon Measurementioning

confidence: 99%

The complete hyperexpansivity analog of the Embry conditions

Athavale

2003

Studia Math.

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Abstract. The Embry conditions are a set of positivity conditions that characterize subnormal operators (on Hilbert spaces) whose theory is closely related to the theory of positive definite functions on the additive semigroup N of non-negative integers. Completely hyperexpansive operators are the negative definite counterpart of subnormal operators. We show that completely hyperexpansive operators are characterized by a set of negativity conditions, which are the natural analog of the Embry conditions for subnormality. While the genesis of the Embry conditions can be traced to the Hausdorff Moment Problem, the genesis of the conditions to be established here lies in the Lévy-Khinchin representation as holding in the context of abelian semigroups. We actually establish the desired negativity criteria for completely hyperexpansive operator tuples.

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“…It is known [22] that there are commuting subnormal operators Sx and S2 such that (Sx, S2) is not a subnormal pair. However there are necessary and sufficient conditions on an «-tuple of commuting subnormal operators for it to be a subnormal «-tuple (see, for example, [3,20]). …”

Section: Preliminariesmentioning

confidence: 99%

Towards a functional calculus for subnormal tuples: the minimal normal extension

Conway¹

1991

Trans. Amer. Math. Soc.

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Abstract. In this paper the study of a functional calculus for subnormal ntuples is initiated and the minimal normal extension problem for this functional calculus is explored. This problem is shown to be equivalent to a mean approximation problem in several complex variables which is solved. An analogous uniform approximation problem is also explored. In addition these general results are applied together with The Area and the The Coarea Formula from Geometric Measure Theory to operators on Bergman spaces and to the tensor product of two subnormal operators. The minimal normal extension of the tensor product of the Bergman shift with itself is completely determined.An «-tuple of commuting operators S = (Sx, ... , Sn) on a Hubert space %? is subnormal if there is an «-tuple N = (Nx, ... , N ) of commuting normal operators on a Hubert space X that contains %? such that for 1 < j < n, Nj£? ç ß? and S¡ = N\ß?. For any commuting «-tuple of operators S there is a notion of spectrum, the Taylor spectrum of S [30] (also see [12]), denoted by er(S). This spectrum is a nonempty compact subset of C". In this general theory it is possible to define (S) for any function analytic in a neighborhood of ct(S) . This functional calculus generalizes the usual Riesz functional calculus for a single operator. When S is assumed to be a subnormal «-tuple, however, a much richer functional calculus is possible. In this case 0(S) can be defined for functions that are weak* limits of analytic functions and 4>(S) becomes a subnormal operator. The central question in this development is "What are the properties of this operator ?" More generally, if S is an «-tuple and l, ... ,

(S) = (x (S), ... , (S)) is a commuting ^-tuple of operators. If S is subnormal, then (S) can be defined for 4> = ( ) consisting of functions that are weak* limits

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Holomorphic Kernels and Commuting Operators

Cited by 22 publications

References 7 publications

Polynomial Optimization on Odd-Dimensional Spheres

Polynomial Optimization on Odd-Dimensional Spheres

The complete hyperexpansivity analog of the Embry conditions

Towards a functional calculus for subnormal tuples: the minimal normal extension

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