A joint spectrum for several commuting operators

Taylor, Joseph L.

doi:10.1016/0022-1236(70)90055-8

Cited by 370 publications

(249 citation statements)

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“…The cohomology spaces of K (T ) are denoted by H p K (T ), 0 ≤ p ≤ n, and their direct sum H K (T ) = n p=0 H p K (T ) is the total cohomology space of K (T ). For more details on this complex and its role in multivariable spectral theory we refer to Taylor [39].…”

Section: Spectral Properties Of Seminormal Systemsmentioning

confidence: 99%

Seminormal systems of operators in Clifford environments

Martin

2017

OpMath

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Abstract. The primary goal of our article is to implement some standard spin geometry techniques related to the study of Dirac and Laplace operators on Dirac vector bundles into the multidimensional theory of Hilbert space operators. The transition from spin geometry to operator theory relies on the use of Clifford environments, which essentially are Clifford algebra augmentations of unital complex C * -algebras that enable one to set up counterparts of the geometric Bochner-Weitzenböck and Bochner-Kodaira-Nakano curvature identities for systems of elements of a C * -algebra. The so derived self-commutator identities in conjunction with Bochner's method provide a natural motivation for the definitions of several types of seminormal systems of operators. As part of their study, we single out certain spectral properties, introduce and analyze a singular integral model that involves Riesz transforms, and prove some self-commutator inequalities.

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Section: Spectral Properties Of Seminormal Systemsmentioning

confidence: 99%

Seminormal systems of operators in Clifford environments

Martin

2017

OpMath

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“…It is shown in [16] that the joint spectrum of a set of commuting operators is nonempty and compact. Thus, when V is invertible, Theorem 3.3 shows that a(W) is nonempty and compact.…”

Section: Proof the Preceding Discussion Has Reduced The Proof Of Thementioning

confidence: 99%

“…The structure of the cohomology module associated with a singular tuple of operators is closely related to the type of singularity of the tuple (see p. 176 of [16]). For instance the elements of the Oth cohomology module correspond to common eigenvectors of the tuple of operators.…”

Section: Proof the Preceding Discussion Has Reduced The Proof Of Thementioning

confidence: 99%

“…When the operators F, are self-adjoint a spectral theorem for the system (1.1) can be obtained from these operators (see [14]). In this paper we will adopt the definition of the joint spectrum of a set of commuting operators given by Taylor in [16] and [17]. An alternative definition of the joint spectrum using commutative Banach algebras is discussed in [6].…”

Section: I) = T-imentioning

confidence: 99%

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Multiparameter spectral theory and Taylor's joint spectrum in Hilbert space

Rynne¹

1988

Proceedings of the Edinburgh Mathematical Society

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Let n≧1 be an integer and suppose that for each i= 1,…,n, we have a Hilbert space Hi and a set of bounded linear operators Ti, Vij:Hi→Hi, j=1,…,n. We define the system of operatorswhere λ=(λ1,…,λn)∈ℂn. Coupled systems of the form (1.1) are called multiparameter systems and the spectral theory of such systems has been studied in many recent papers. Most of the literature on multiparameter theory deals with the case where the operators Ti and Vij are self-adjoint (see [14]). The non self-adjoint case, which has received relatively little attention, is discussed in [12] and [13].

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“…For rational f=g/h (g and h polynomials), one expects such a result to hold provided/^) is defined; one expects, accordingly, to assume h has no zeros on the joint spectrum of A. Several definitions of joint spectrum recommend themselves [11], but this does not concern us here. The formally weakest restriction will be to exclude zeros of h from the smallest possible set.…”

Section: This In Turn Is ^F(a V (A) 9 a V (B)) Just Because A^ab)^ Amentioning

confidence: 99%