Vladimír Müller

Spectral theory of linear operators : and spectral systems in Banach algebras

Preface .- I. Banach Algebras .- II. Operators .- III. Essential Spectrum .- IV. Taylor Spectrum .- V. Orbits and Capacity .- Appendix .- Bibliography

Standard models for some commuting multioperators

An analogue to the Sz.-Nagy-Foia5 dilation theory is presented for several commuting operators on a Hilbert space. The dilation theory of Hilbert space contractions, a reference of which is the excellent monograph [9], has not yet found a complete counterpart valid for several (commuting) Hilbert space operators. Apart from the results of this type already contained in [9] and other results of general character (i.e., Arvesons or Stinesprings extension theorem), there are not too many contributions specialized to the multiparameter dilation theory. Among these contributions, we quote the works [7, 3-5, 10], which are closer to our topics. The aim of this paper is to analyse some positivity conditions for commuting multioperators, which ensure the unitary equivalence of these objects to some standard models consisting of backwards multishifts. We emphasize, in particular, a situation that seems to satisfy the requirements of an appropriate extension for the dilation theory of Hilbert space contractions and that exploits the results already obtained in [4, 10]. Our finer methods also imply better statements of some assertions from [10] and provide an answer to Question 4.5 from [5]. Let n > 1 be an integer, and let Zn be the set of all n-tuples of nonnegative integers (i.e., the multi-indices of length n). If a = (a, ...I, an) E Zn+, we set, as usual, lal =a,+ + -+a, and a! =a1! .. !. For a,ficZ+ wewrite a + f = (a1 + f1j ... , an + fin), and a < fi whenever aci < /i (i = 1, ... , n). Let H be a Hilbert space, and let 2(H) be the algebra of all bounded linear operators on H. An element T = (T,, ... , T,) E 2(H)n such that T,, ... , T, mutually commute will be designated as a commuting multioperator (briefly, a c.m.). Let T E Y(H)n be a c.m. We define the operator MT: Y(H) -* J(H) by the formula

The edge reconstruction hypothesis is true for graphs with more than n · log2n edges

Abstract It is shown that a graph with n vertices and more than n · log 2 n edges can be uniquely reconstructed from its edge-deleted subgraphs.

Either tournaments or algebras

The paper deals with tournaments (i.e., with trichotomic relations) and their homomorphisms. The study of tournaments by means of their homomorphisms is natural as tournaments are algebras of a special kind. We prove (1) theorems which relate combinatorial and algebraic notions (e.g., the score of a tournament and the monoid of its endomorphisms); (2) theorems concerned with strictly algebraic aspects of tournaments (e.g., characterizing the lattice of congruences of a tournament). Our main result is that the group of automorphisms and the lattice of congruences of a tournament are in general independent. In the last part of the paper we give some examples and applications to other fields.

The rate of convergence in the method of alternating projections

A generalization of the cosine of the Friedrichs angle between two subspaces to a parameter associated to several closed subspaces of a Hilbert space is given. This parameter is used to analyze the rate of convergence in the von Neumann-Halperin method of cyclic alternating projections. General dichotomy theorems are proved, in the Hilbert or Banach space situation, providing conditions under which the alternative QUC/ASC (quick uniform convergence versus arbitrarily slow convergence) holds. Several meanings for ASC are proposed.

On colorings of graphs without short cycles

Abstract The following theorem is proved: Let n, k be natural numbers, n ⩾ 3, let A be a set and A 1 , A 2 ,…, A r different decompositions of A into at most n classes. Then there exists an n-chromatic graph G = 〈 V ( G ), E ( G )〉 such that A ⊂ V ( G ), G does not contain cycles of length ⩽ k and G has just r colorings B 1 ,…, B r by n colours such that A i = B i |A, i = 1,…,r. From this theorem follows immediately existence of uniquely colorable graphs without short cycles. Further, characterizations of subgraphs of critical graphs may be given.

Jauch-Piron states on concrete quantum logics

We exhibit an example of a concrete (=set-representable) quantum logic which is not a Boolean algebra such that every state on it is Jauch-Piron. This gives a negative answer to a problem raised by Navara and Pták. Further we show that such an example does not exist in the class of complete (i.e., closed under arbitrary disjoint unions) concrete logics.

Power bounded operators and supercyclic vectors

We show that each power bounded operator with spectral radius equal to one on a reflexive Banach space has a nonzero vector which is not supercyclic. Equivalently, the operator has a nontrivial closed invariant homogeneous subset. Moreover, the operator has a nontrivial closed invariant cone if 1 belongs to its spectrum. This generalizes the corresponding results for Hilbert space operators. For non-reflexive Banach spaces these results remain true: however, the non-supercyclic vector (invariant cone, respectively) relates to the adjoint of the operator.

Orbits, weak orbits and local capacity of operators

LetT be an operator on a Banach spaceX. We give a survey of results concerning orbits {Tnx:n=0,1,...} and weak orbits {〈Tnx,x*〉:n=0,1,...} ofT wherex∈X andx*∈X*. Further we study the local capacity of operators and prove that there is a residual set of pointsx∈X with the property that the local capacity cap(T, x) is equal to the global capacity capT. This is an analogy to the corresponding result for the local spectral radius.

Concrete quantum logics with covering properties

LetL be a concrete (=set-representable) quantum logic. Letn be a natural number (or, more generally, a cardinal). We say thatL admits intrinsic coverings of the ordern, and writeL∈Cn, if for any pairA, B∈L we can find a collection {Ci∶ i∈I}, where cardI<n andCi∈L for anyi∈I, such thatA ∩B=∪i∈lCi. Thus, in a certain sense, ifL∈Cn, then “the rate of noncompatibility” of an arbitrary pairA,B∈L is less than a given numbern. In this paper we first consider general and combinatorial properties of logics ofCn and exhibit typical examples. In particular, for a givenn we construct examples ofL∈Cn+1\Cn. Further, we discuss the relation of the classesCn to other classes of logics important within the quantum theories (e.g., we discover the interesting relation to the class of logics which have an abundance of Jauch-Piron states). We then consider conditions on which a class of concrete logics reduce to Boolean algebras. We conclude with some open questions.