Multiparameter problems and joint spectra

1982 ◽  
Vol 93 (1-2) ◽  
pp. 129-135 ◽  
Author(s):  
D. F. McGhee
Keyword(s):  
Bounded Operators ◽  
Joint Spectrum ◽  
Intimate Connection ◽  
Commuting Operators ◽  
Multiparameter Problem ◽  
Adjoint Operators

SynopsisIn this paper, we demonstrate an intimate connection between the spectrum of a multiparameter problem and the joint spectrum of an associated set of commuting operators, and show that the spectrum of a multiparameter problem involving bounded operators is non-empty. Multiparameter systems involving compact and self-adjoint operators are considered, and some simplification of results in the literature are noted.

On the Right Hamiltonian for Singular Perturbations: General Theory

1997 ◽  
Vol 09 (05) ◽  
pp. 609-633 ◽  
Author(s):  
Hagen Neidhardt ◽  
Valentin Zagrebnov
Keyword(s):  
General Theory ◽  
Symmetric Operator ◽  
Stability Domain ◽  
Bounded Operators ◽  
Friedrichs Extension ◽  
Abstract Problem ◽  
Dense Domain ◽  
Adjoint Extension ◽  
The Right ◽  
Adjoint Operators

Let the pair of self-adjoint operators {A≥0,W≤0} be such that: (a) there is a dense domain [Formula: see text] such that [Formula: see text] is semibounded from below (stability domain), (b) the symmetric operator [Formula: see text] is not essentially self-adjoint (singularity of the perturbation), (c) the Friedrichs extension [Formula: see text] of [Formula: see text] is maximal with respect to W, i.e., [Formula: see text]. [Formula: see text]. Let [Formula: see text] be a regularizing sequence of bounded operators which tends in the strong resolvent sense to W. The abstract problem of the right Hamiltonian is: (i) to give conditions such that the limit H of self-adjoint regularized Hamiltonians [Formula: see text] exists and is unique for any self-adjoint extension [Formula: see text] of [Formula: see text], (ii) to describe the limit H. We show that under the conditions (a)–(c) there is a regularizing sequence [Formula: see text] such that [Formula: see text] tends in the strong resolvent sense to unique (right Hamiltonian) [Formula: see text], otherwise the limit is not unique.

Monotone convergence theorems for semi-bounded operators and forms with applications

2010 ◽  
Vol 140 (5) ◽  
pp. 927-951 ◽  
Author(s):  
Jussi Behrndt ◽  
Seppo Hassi ◽  
Henk de Snoo ◽  
Rudi Wietsma
Keyword(s):  
Differential Operators ◽  
Multiplication Operators ◽  
Monotone Convergence ◽  
Monotone Sequence ◽  
Bounded Operators ◽  
Von Neumann ◽  
Sturm Liouville ◽  
Adjoint Operators ◽  
Liouville Operators

AbstractLet Hn be a monotone sequence of non-negative self-adjoint operators or relations in a Hilbert space. Then there exists a self-adjoint relation H∞ such that Hn converges to H∞ in the strong resolvent sense. This result and related limit results are explored in detail and new simple proofs are presented. The corresponding statements for monotone sequences of semi-bounded closed forms are established as immediate consequences. Applications and examples, illustrating the general results, include sequences of multiplication operators, Sturm–Liouville operators with increasing potentials, forms associated with Kreĭn–Feller differential operators, singular perturbations of non-negative self-adjoint operators and the characterization of the Friedrichs and Kreĭn–von Neumann extensions of a non-negative operator or relation.

scholarly journals Simultaneous Upper Triangular Forms for Commuting Operators in a Finite von Neumann Algebra

2019 ◽  
Vol 72 (5) ◽  
pp. 1188-1245
Author(s):  
Ian Charlesworth ◽  
Ken Dykema ◽  
Fedor Sukochev ◽  
Dmitriy Zanin
Keyword(s):  
Functional Calculus ◽  
Von Neumann Algebra ◽  
Von Neumann ◽  
Holomorphic Functional Calculus ◽  
Joint Spectrum ◽  
Natural Properties ◽  
Tracial State ◽  
Commuting Operators ◽  
Finite Von Neumann Algebra

AbstractThe joint Brown measure and joint Haagerup–Schultz projections for tuples of commuting operators in a von Neumann algebra equipped with a faithful tracial state are investigated, and several natural properties are proved for these. It is shown that the support of the joint Brown measure is contained in the Taylor joint spectrum of the tuple, and also in the ostensibly smaller left Harte spectrum. A simultaneous upper triangularization result for finite commuting tuples is proved, and the joint Brown measure and joint Haagerup–Schultz projections are shown to behave well under the Arens multivariate holomorphic functional calculus of such a commuting tuple.

scholarly journals Joint spectra and joint numerical ranges for pairwise commuting operators in Banach spaces

1988 ◽  
Vol 30 (2) ◽  
pp. 145-153 ◽  
Author(s):  
Volker Wrobel
Keyword(s):  
Banach Space ◽  
Convex Hull ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Numerical Range ◽  
Linear Operators ◽  
Continuous Linear ◽  
Joint Spectrum ◽  
Commuting Operators ◽  
Continuous Linear Operators

In a recent paper M. Cho [5] asked whether Taylor's joint spectrum σ(a1, …, an; X) of a commuting n-tuple (a1,…, an) of continuous linear operators in a Banach space X is contained in the closure V(a1, …, an; X)- of the joint spatial numerical range of (a1, …, an). Among other things we prove that even the convex hull of the classical joint spectrum Sp(a1, …, an; 〈a1, …, an〉), considered in the Banach algebra 〈a1, …, an〉, generated by a1, …, an, is contained in V(a1, …, an; X)-.

scholarly journals On the Self-Adjointness of H+A∗+A

2020 ◽  
Vol 23 (4) ◽  
Author(s):  
Andrea Posilicano
Keyword(s):  
Field Theory ◽  
Annihilation Operator ◽  
The Self ◽  
Quantum Field ◽  
Bounded Operators ◽  
Nelson Model ◽  
Adjoint Operators ◽  
Nonperturbative Theory

AbstractLet $H:\text {dom}(H)\subseteq \mathfrak {F}\to \mathfrak {F}$ H : dom ( H ) ⊆ F → F be self-adjoint and let $A:\text {dom}(H)\to \mathfrak {F}$ A : dom ( H ) → F (playing the role of the annihilation operator) be H-bounded. Assuming some additional hypotheses on A (so that the creation operator A∗ is a singular perturbation of H), by a twofold application of a resolvent Kreı̆n-type formula, we build self-adjoint realizations $\widehat H$ H ̂ of the formal Hamiltonian H + A∗ + A with $\text {dom}(H)\cap \text {dom}(\widehat H)=\{0\}$ dom ( H ) ∩ dom ( H ̂ ) = { 0 } . We give an explicit characterization of $\text {dom}(\widehat H)$ dom ( H ̂ ) and provide a formula for the resolvent difference $(-\widehat H+z)^{-1}-(-H+z)^{-1}$ ( − H ̂ + z ) − 1 − ( − H + z ) − 1 . Moreover, we consider the problem of the description of $\widehat H$ H ̂ as a (norm resolvent) limit of sequences of the kind $H+A^{*}_{n}+A_{n}+E_{n}$ H + A n ∗ + A n + E n , where the An’s are regularized operators approximating A and the En’s are suitable renormalizing bounded operators. These results show the connection between the construction of singular perturbations of self-adjoint operators by Kreı̆n’s resolvent formula and nonperturbative theory of renormalizable models in Quantum Field Theory; in particular, as an explicit example, we consider the Nelson model.

scholarly journals Joint spectra for commuting operators

1985 ◽  
Vol 28 (2) ◽  
pp. 233-248 ◽  
Author(s):  
A. KällstrÖm ◽  
B.D. Sleeman
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Chain Complex ◽  
Joint Spectrum ◽  
Commuting Operators ◽  
A Chain

The theory of joint spectra for commuting operators in a Hilbert space has recently been studied by several authors (Vasilescu [11,12], Curto [4,5], and Cho-Takaguchi[2,3]). In this paper we willuse the definition by Taylor [10] of the joint spectrum to show that thejoint spectrum is determined by the action of certain "Laplacians"(cf. Curto [4,5]) of a chain-complex of Hilbert spaces.

scholarly journals On Differences of Unitarily Equivalent Self-Adjoint Operators

1960 ◽  
Vol 4 (3) ◽  
pp. 103-107 ◽  
Author(s):  
C. R. Putnam
Keyword(s):  
Continuous Operator ◽  
Bounded Operators ◽  
Constant Multiple ◽  
Additional Hypothesis ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case ◽  
The Difference ◽  
Image Position ◽  
Adjoint Operators

1. All operators considered in this paper are bounded operators on a Hilbert space. In case A and B are self-adjoint, certain conditions on A, B and their differenceassuring the unitary equivalence of Aand B,have recently been obtained by Rosenblum [6] and Kato [2]. The present paper will consider the problem of investigating consequences of an assumed relation of type (2) for some unitary U together with an additional hypothesis that the difference H of (1) be non-negative, so thatFirst, it is easy to see that if only (2) and (3) are assumed, thereby allowing H = 0, relation (2) can hold for A arbitrary with U = I (identity) and B = A. If H = 0 in (3) is not allowed, however (an impossible assumption in the finite dimensional case, incidentally, since then the trace of H is zero and hence H = 0), it will be shown, among other things, that any unitary operator U for which (2) and (3) hold must have a spectrum with a positive measure (as a consequence of (i) of Theorem 2 below). Moreover A (hence B) cannot differ from a completely continuous operator by a constant multiple of the identity (Theorem 1). In case 0 is not in the point spectrum of H, then U is even absolutely continuous (see (iv) of Theorem 2). In § 4, applications to semi-normal operators will be given.

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