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Commutativity of compact selfadjoint operators

G. Greiner, W. Ricker (1995)

Studia Mathematica

The relationship between the joint spectrum γ(A) of an n-tuple A = ( A 1 , . . . , A n ) of selfadjoint operators and the support of the corresponding Weyl calculus T(A) : f ↦ f(A) is discussed. It is shown that one always has γ(A) ⊂ supp (T(A)). Moreover, when the operators are compact, equality occurs if and only if the operators A j mutually commute. In the non-commuting case the equality fails badly: While γ(A) is countable, supp(T(A)) has to be an uncountable set. An example is given showing that, for non-compact operators,...

Exponential bounds for noncommuting systems of matrices

Brian Jefferies (2001)

Studia Mathematica

It is shown that a finite system T of matrices whose real linear combinations have real spectrum satisfies a bound of the form | | e i T , ζ | | C ( 1 + | ζ | ) s e r | ζ | . The proof appeals to the monogenic functional calculus.

Fourier-like methods for equations with separable variables

Danuta Przeworska-Rolewicz (2009)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

It is well known that a power of a right invertible operator is again right invertible, as well as a polynomial in a right invertible operator under appropriate assumptions. However, a linear combination of right invertible operators (in particular, their sum and/or difference) in general is not right invertible. It will be shown how to solve equations with linear combinations of right invertible operators in commutative algebras using properties of logarithmic and antilogarithmic mappings. The...

Function theory in sectors

Brian Jefferies (2004)

Studia Mathematica

It is shown that there is a one-to-one correspondence between uniformly bounded holomorphic functions of n complex variables in sectors of ℂⁿ, and uniformly bounded functions of n+1 real variables in sectors of n + 1 that are monogenic functions in the sense of Clifford analysis. The result is applied to the construction of functional calculi for n commuting operators, including the example of differentiation operators on a Lipschitz surface in n + 1 .

In search of the invisible spectrum

Nikolai Nikolski (1999)

Annales de l'institut Fourier

In this paper, we begin the study of the phenomenon of the “invisible spectrum” for commutative Banach algebras. Function algebras, formal power series and operator algebras will be considered. A quantitative treatment of the famous Wiener-Pitt-Sreider phenomenon for measure algebras on locally compact abelian (LCA) groups is given. Also, our approach includes efficient sharp estimates for resolvents and solutions of higher Bezout equations in terms of their spectral bounds. The smallest “spectral...

On some constructions of new triangular norms.

Radko Mesiar (1995)

Mathware and Soft Computing

We discuss the properties of two types of construction of a new t-norm from a given t-norm proposed recently by B. Demant, namely the dilatation and the contraction. In general, the dilatation of a t-norm is an ordinal sum t-norm and the continuity of the outgoing t-norm is preserved. On the other hand, the contraction may violate the continuity as well as the non-continuity of the outgoing t-norm. Several examples are given.

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