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On a binary relation between normal operators

Takateru Okayasu, Jan Stochel, Yasunori Ueda (2011)

Studia Mathematica

The main goal of this paper is to clarify the antisymmetric nature of a binary relation ≪ which is defined for normal operators A and B by: A ≪ B if there exists an operator T such that E A ( Δ ) ≤ T * E B ( Δ ) T for all Borel subset Δ of the complex plane ℂ, where E A and E B are spectral measures of A and B, respectively (the operators A and B are allowed to act in different complex Hilbert spaces). It is proved that if A ≪ B and B ≪ A, then A and B are unitarily equivalent, which shows that the relation ≪ is a partial order...

On a function that realizes the maximal spectral type

Krzysztof Frączek (1997)

Studia Mathematica

We show that for a unitary operator U on L 2 ( X , μ ) , where X is a compact manifold of class C r , r ∈ ℕ ∪ ∞ , ω , and μ is a finite Borel measure on X, there exists a C r function that realizes the maximal spectral type of U.

On a generalization of Lumer-Phillips' theorem for dissipative operators in a Banach space

Driss Drissi (1998)

Studia Mathematica

Using [1], which is a local generalization of Gelfand's result for powerbounded operators, we first give a quantitative local extension of Lumer-Philips' result that states conditions under which a quasi-nilpotent dissipative operator vanishes. Secondly, we also improve Lumer-Phillips' theorem on strongly continuous semigroups of contraction operators.

On a theorem of Gelfand and its local generalizations

Driss Drissi (1997)

Studia Mathematica

In 1941, I. Gelfand proved that if a is a doubly power-bounded element of a Banach algebra A such that Sp(a) = 1, then a = 1. In [4], this result has been extended locally to a larger class of operators. In this note, we first give some quantitative local extensions of Gelfand-Hille’s results. Secondly, using the Bernstein inequality for multivariable functions, we give short and elementary proofs of two extensions of Gelfand’s theorem for m commuting bounded operators, T 1 , . . . , T m , on a Banach space X.

On a Weyl-von Neumann type theorem for antilinear self-adjoint operators

Santtu Ruotsalainen (2012)

Studia Mathematica

Antilinear operators on a complex Hilbert space arise in various contexts in mathematical physics. In this paper, an analogue of the Weyl-von Neumann theorem for antilinear self-adjoint operators is proved, i.e. that an antilinear self-adjoint operator is the sum of a diagonalizable operator and of a compact operator with arbitrarily small Schatten p-norm. On the way, we discuss conjugations and their properties. A spectral integral representation for antilinear self-adjoint operators is constructed....

On limits of L p -norms of an integral operator

Pavel Stavinoha (1994)

Applications of Mathematics

A recurrence relation for the computation of the L p -norms of an Hermitian Fredholm integral operator is derived and an expression giving approximately the number of eigenvalues which in absolute value are equal to the spectral radius is determined. Using the L p -norms for the approximation of the spectral radius of this operator an a priori and an a posteriori bound for the error are obtained. Some properties of the a posteriori bound are discussed.

On product of projections

Mohammad Sal Moslehian (2004)

Archivum Mathematicum

An operator with infinite dimensional kernel is positive iff it is a positive scalar times a certain product of three projections.

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