Operants: A Functional Calculus for Non-Commuting Operators

1970 ◽  
pp. 172-187 ◽  
Author(s):  
Edward Nelson
Keyword(s):  
Functional Calculus ◽  
Commuting Operators

scholarly journals On non-holomorphic functional calculus for commuting operators

2003 ◽  
Vol 93 (1) ◽  
pp. 109 ◽  
Author(s):  
Sebastian Sandberg
Keyword(s):  
Cohomology Class ◽  
Functional Calculus ◽  
General Scheme ◽  
Holomorphic Functions ◽  
Holomorphic Functional Calculus ◽  
Resolvent Identity ◽  
Commuting Operators

We provide a general scheme to extend Taylor's holomorphic functional calculus for several commuting operators to classes of non-holomorphic functions. These classes of functions will depend on the growth of the operator valued forms that define the resolvent cohomology class. The proofs are based on a generalization of the so-called resolvent identity to several commuting operators.

scholarly journals Discrete quadratic estimates and holomorphic functional calculi in Banach spaces

1998 ◽  
Vol 58 (2) ◽  
pp. 271-290 ◽  
Author(s):  
Edwin Franks ◽  
Alan McIntosh
Keyword(s):  
Functional Calculus ◽  
Sufficient Conditions ◽  
Uniform Norm ◽  
Commuting Operators ◽  
Orthonormal Sequence ◽  
Quadratic Estimates ◽  
Holomorphic Functional Calculi ◽  
Necessary And Sufficient ◽  
Dyadic Decomposition ◽  
Uniformly Bounded

We develop a discrete version of the weak quadratic estimates for operators of type w explained by Cowling, Doust, McIntosh and Yagi, and show that analogous theorems hold. The method is direct and can be generalised to the case of finding necessary and sufficient conditions for an operator T to have a bounded functional calculus on a domain which touches σ(T) nontangentially at several points. For operators on Lp, 1 < p < ∞, it follows that T has a bounded functional calculus if and only if T satisfies discrete quadratic estimates. Using this, one easily obtains Albrecht's extension to a joint functional calculus for several commuting operators. In Hilbert space the methods show that an operator with a bounded functional calculus has a uniformly bounded matricial functional calculus.The basic idea is to take a dyadic decomposition of the boundary of a sector Sv. Then on the kth ingerval consider an orthonormal sequence of polynomials . For h ∈ H∞(Sν), estimates for the uniform norm of h on a smaller sector Sμ are obtained from the coefficients akj = (h, ek, j). These estimates are then used to prove the theorems.

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 A functional calculus for sesquihermitian operators on quaternionic Hilbert space

1976 ◽  
Vol 21 (1) ◽  
pp. 96-107
Author(s):  
N. C. Powers
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Spectral Distribution ◽  
Functional Calculus ◽  
Complex Hilbert Space ◽  
Linear Mapping ◽  
Linear Operators ◽  
Commuting Operators ◽  
Real Linear ◽  
Quaternionic Hilbert Space

AbstractA continuous real-linear operator A = A0 + i1A1 + i2A2 + i3A3 on a quaternionic Hilbert space is called sesquihermitian if the linear operators Av are Hermitian; this condition is independent of the choice of quaternion basis (i1,i2,i3). The joint spectral distribution of the Av provides a functional calculus for sesquihermitian operators and real-valued C∞-functions on . This calculus is independent of the quaternion basis and extends naturally to quaternion-valued functions to give a continuous quaternion-linear mapping from the algebra of these functions to that of sesquihermitian operators. The mapping is not, in general, multiplicative unless the Av commute, in which case it agrees with that for several commuting operators on complex Hilbert space.

The -spectrum and the -functional calculus

2012 ◽  
Vol 142 (3) ◽  
pp. 479-500 ◽  
Author(s):  
Fabrizio Colombo ◽  
Irene Sabadini
Keyword(s):  
Banach Space ◽  
Functional Calculus ◽  
Resolvent Operator ◽  
Monogenic Functions ◽  
Commuting Operators ◽  
Correct Definition ◽  
Slice Monogenic Functions ◽  
Definition Of ◽  
Integral Version

In some recent papers (called $\mathcal{S}$-functional calculus) for n-tuples of both bounded and unbounded not-necessarily commuting operators. The $\mathcal{S}$-functional calculus is based on the notion of $\mathcal{S}$-spectrum, which naturally arises from the definition of the $\mathcal{S}$-resolvent operator for n-tuples of operators. The $\mathcal{S}$-resolvent operator plays the same role as the usual resolvent operator for the Riesz–Dunford functional calculus, which is associated to a complex operator acting on a Banach space. When one considers commuting operators (bounded or unbounded) there is the possibility of simplifying the computation of the $\mathcal{S}$-spectrum. In fact, in this case we can use the F-spectrum, which is easier to compute than the $\mathcal{S}$-spectrum. In the case of commuting operators, our functional calculus is based on the $\mathcal{F}$-spectrum and will be called $\mathcal{SC}$-functional calculus. We point out that for a correct definition of the $\mathcal{S}$-resolvent operator and of the $\mathcal{SC}$-resolvent operator in the unbounded case we have to face different extension problems. Another reason for a more detailed study of the $\mathcal{F}$-spectrum is that it is related to the $\mathcal{F}$-functional calculus which is based on the integral version of the Fueter mapping theorem. This functional calculus is associated to monogenic functions constructed by starting from slice monogenic functions.

scholarly journals Holomorphic functional calculi and sums of commuting opertors

1998 ◽  
Vol 58 (2) ◽  
pp. 291-305 ◽  
Author(s):  
David Albrecht ◽  
Edwin Franks ◽  
Alan McIntosh
Keyword(s):  
Banach Space ◽  
Holomorphic Function ◽  
Functional Calculus ◽  
Holomorphic Functional Calculus ◽  
Regularity Problem ◽  
Commuting Operators ◽  
Consistent Manner ◽  
Holomorphic Functional Calculi ◽  
The Individual

Let S and T be commuting operators of type ω and type ϖ in a Banach space X. Then the pair has a joint holomorphic functional calculus in the sense that it is possible to define operators f(S, T) in a consistent manner, when f is a suitable holomorphic function defined on a product of sectors. In particular, this gives a way to define the sum S + T when ω + ϖ < π. We show that this operator is always of type μ where μ = max{ω, ϖ}. We explore when bounds on the individual functional calculi of S and T imply bounds on the functional calculus of the pair (S, T), and some implications for the regularity problem of when ∥(S + T)u∥ is equivalent to ∥Su∥ + ∥Tu∥.

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