ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. An algebra is called uniform if all classes of any of its congruences are of the same size. We prove that every finite uniform lattice is congruence permutable. We also show that this result extends neither to arbitrary finite algebras nor to infinite lattices.

AMS Subject Classification (1991): 06B10

Received November 18, 2004. (Registered under 5881/2009.)

Abstract. We calculate the number of unary clones (submonoids of the full transformation monoid) containing the permutations, on an infinite base set. It turns out that this number is quite large, on some cardinals as large as the whole clone lattice. Moreover we find that, with one exception, even the cardinalities of the intervals between the monoid of all permutations and the maximal submonoids of the full transformation monoid are as large. Whether or not the only exception is of the same cardinality as the other intervals depends on additional axioms of set theory.

AMS Subject Classification (1991): 08A40, 08A05

Keyword(s): clone lattice, permutations, unary clones, transformation monoid, submonoids

Received October 20, 2004, and in revised form May 20, 2005. (Registered under 5882/2009.)

Abstract. We prove that every three-element Mal'tsev algebra is determined, up to term equivalence, by its subalgebras, congruences, internal homomorphisms, labels of prime quotients (in the sense of tame congruence theory), and in certain special cases, by subalgebras of its cube. Thus there exist only finitely many three-element Mal'tsev algebras, up to term equivalence.

AMS Subject Classification (1991): 08A40, 08B05, 08A20

Received December 21, 2000, and in final form April 20, 2005. (Registered under 5883/2009.)

Abstract. For any integer $a$, let $G(a)=0$ if $2|a$ and $G(a)=1$ if $2\not|a$. It is proved that for any two subsets $A$ and $B$ of $\{1, 2,\ldots, n\} $ with $|A|+|B|+G(|A|+|B|)\ge(4n+5)/3$, there exist at least $|A|+|B|+G(|A|+|B|)-3$ consecutive integers in the restricted sumset $A\hat{+} B=\{a+b : a\in A, b\in B, a\not= b\} $. Furthermore, it is proved that for any two subsets $A$ and $B$ of $\{1, 2,\ldots, n\} $ with $|A|+|B|+G(|A|+|B|)\ge(4n+4)/3$, there exists an arithmetic progression in $A\hat{+} B$ with length at least $n-1$; for any integer $r$ with $8\le r+G(r)< (4n+4)/3$, there exist two subsets $A$ and $B$ of $\{1, 2,\ldots, n\} $ with $|A|+|B|=r$ such that each arithmetic progression in $A\hat{+} B$ has length at most $2(n+1)/3$.

AMS Subject Classification (1991): 11B75

Received December 30, 2002, and in revised form August 26, 2005. (Registered under 5884/2009.)

Abstract. It is well-known that an inverse monoid is factorizable if and only if it is a homomorphic image of a semidirect product of a semilattice (with identity) by a group. We use this structure to describe a presentation of an arbitrary factorizable inverse monoid in terms of presentations of its group of units and semilattice of idempotents, together with some other data. We apply this theory to quickly deduce a well-known presentation of the symmetric inverse monoid on a finite set.

AMS Subject Classification (1991): 20M05, 20M18; 20M20

Keyword(s): Factorizable inverse monoid, presentations, symmetric inverse monoid

Received July 20, 2004, and in revised form June 10, 2005. (Registered under 5885/2009.)

Abstract. The ${\cal R}$-compatible semigroup varieties are classified and described. It is shown that a periodic semigroup variety contains at most three maximal ${\cal R}$-compatible subvarieties, and each ${\cal R}$-compatible subvariety is contained in one of the maximal ones. The semigroup varieties which are minimal for not being ${\cal R}$-compatible are found: they are countably infinite in number. There are three maximal ${\cal R}$-compatible pseudovarieties of semigroups. Analogues for varieties and pseudovarieties of monoids are established. If an ${\cal R}$-compatible monoid variety contains a nonabelian group, then this variety is periodic and consists of completely regular monoids only.

AMS Subject Classification (1991): 20M07, 20M10

Received December 9, 2004, and in revised form August 16, 2005. (Registered under 5886/2009.)

Abstract. The Twin Dragon and Rauzy fractals are intersected with the real axis. In the Twin Dragon case, unexpectedly from its fractal nature, the intersection is an interval characterized by a finite automaton. For the case of the Rauzy fractal, it is proved that the intersection has infinitely many components.

AMS Subject Classification (1991): 11K26, 11A63, 37B50, 52C23, 28A80

Keyword(s): Twin Dragon, Rauzy Fractal, Tiling

Received March 9, 2004, and in revised form July 14, 2005. (Registered under 5887/2009.)

Abstract. We consider semilinear elliptic equations with nonsmooth potential and resonant at high parts of the spectrum of $\left(-\Delta,H^1_0(Z)\right )$. Asymptotically at infinity we permit double resonance of ${\partial j(z,\zeta )\over\zeta }$ between two consecutive eigenvalues. The resonance is complete in the higher part of the spectrum, incomplete in the lower part. We also permit resonance asymptotically at zero. Using a variational approach based on nonsmooth critical point theory, we prove the existence of a nontrivial solution.

AMS Subject Classification (1991): 35J20, 35J85

Keyword(s): Eigenvalues, resonance, unique continuation property, orthogonality, locally Lipschitz function, Clarke subdifferential, nonsmooth critical point theory, nonsmooth linking theorem

Received November 24, 2003. (Registered under 5888/2009.)

Abstract. Using the technique of minimal vectors, as introduced by Ansari and Enflo and generalized by Troitsky, we provide new results concerning the existence of invariant subspaces for operators satisfying conditions similar to those of Lomonosov's theorems. Most of the operators to which the theory applies are quasinilpotent, but a wide class of weighted shifts, including vectorial shifts, is studied as a further application.

AMS Subject Classification (1991): 41A29, 47A15, 47B37, 46B10

Keyword(s): Minimal vectors, Hyperinvariant subspaces, Weighted shifts, Quasinilpotent operators

Received September 6, 2004, and in final form June 1, 2005. (Registered under 5889/2009.)

Abstract. We prove two general theorems of embedding type and give a short survey on the previous results to be improved. It is also shown that one of the embedding relations cannot be weakened in general.

AMS Subject Classification (1991): 42A32, 42A10

Received January 11, 2005. (Registered under 5890/2009.)

Abstract. We present a new proof of the following theorem. There exists an orthonormal system $(\Phi_n)_{n\geq1}$ in ${\msbm L}_2(0,1)$, such that corresponding to each sequence $(w_n)_{n\geq1}$ of positive numbers, $w_n =o(\log_2^2n)$ as $n\to\infty $, there is a series $$\sum_{n\geq1} a_n\Phi_n(x)$$ that diverges a.e. and whose coefficients satisfy $$\sum_{n\geq1} a_n^2 w_n < \infty.$$ Our proof {\it does not} depend on the properties of the Hilbert matrix $({1\over i-j})_{i,j\geq1, i\not=j}.$ Possible simplifications of the proof of Tandori's theorem are also discussed. More precisely, we give a new proof of the famous lemma of Menshov being a starting point of a theory of {\it divergence} of orthogonal series.

AMS Subject Classification (1991): 42C15

Keyword(s): orthogonal series, Rademacher-Menshov theorem, Menshov lemma

Received January 19, 2004, and in revised form August 2, 2005. (Registered under 5891/2009.)

Abstract. We represent a $C^*$-algebra generated by partial isometries having commuting range and support projections as the quotient of a partial crossed product of an abelian $C^*$-algebra and a free group. In particular, we get such representations for certain Cuntz--Krieger type algebras. Under special conditions the quotient can be represented directly as a partial crossed product of an abelian $C^*$-algebra and a quotient of the free group.

AMS Subject Classification (1991): 46L05, 46L55

Received June 16, 2004, and in revised form August 2, 2005. (Registered under 5892/2009.)

Abstract. A Hilbert space bounded linear operator is said to be regular if its range is closed and its kernel is included in the intersection of the ranges of all iterates. We prove that if $T$ is regular, then $T$ is similar to a partial isometry if and only if $T$ is power bounded and there exists a power bounded operator $S$ such that $TST =T$. This is a generalization of a similarity criterion due to B. Sz.-Nagy. The regularity condition cannot be avoided. Indeed, using Pisier's example of a polynomially bounded operator not similar to a contraction, two polynomially bounded operators $T$ and $S$ are constructed such that $TST = T$ and $STS =S$, but $T$ is not similar to a partial isometry.

AMS Subject Classification (1991): 47A05, 47A10, 47B47

Keyword(s): partial isometries, similarity problems, generalized inverse, generalized spectrum, the commutator equation

Received April 28, 2005, and in final form September 14, 2005. (Registered under 5893/2009.)

Abstract. We describe local spectral properties of upper triangular block operators having a diagonal that satisfies the single valued extension property together with its adjoint. Some results on $n$-normal operators and Jordan operators are recaptured.

AMS Subject Classification (1991): 47A 11, 47A10

Keyword(s): local spectral theory, operator matrices, spectra

Received November 5, 2004, and in revised form August 2, 2005. (Registered under 5894/2009.)

Abstract. Multivariable extensions of the Scott Brown technique are used to prove that on a quite general class of domains $D$ in ${\msbm C}^n$, or in a complex submanifold of ${\msbm C}^n$, each subnormal tuple $T\in L(H)^n$ with an isometric weak$^*$ continuous $H^{\infty }(D)$-functional calculus is {\it reflexive}. To obtain our results we use methods developed by Aleksandrov in his abstract approach to the inner function problem, and we prove new results for Henkin measures on suitable complex domains.

AMS Subject Classification (1991): 47A13, 47A15, 47A60, 47B20, 47L45

Received December 28, 2004. (Registered under 5895/2009.)

Abstract. In this note we give several examples of invertible operators $T$ on Hilbert space such that the sets ${\cal C}(T)$ and ${\cal C}(T^{-1})$ of cyclic vectors for $T$ and $T^{-1}$, respectively, are different. This forecloses one possible approach to solving the famous problem of Halmos: if $T$ has a nontrivial invariant subspace, then does necessarily $T^{-1}$ have one too?

AMS Subject Classification (1991): 47A15

Keyword(s): Invariant subspaces

Received May 4, 2005, and in revised form September 20, 2005. (Registered under 5896/2009.)

Abstract. There exists a unitary operator $V$ in $L_2[0, 1)$ such that, for any increasing sequence of indices $n(k)$, the averages ${1\over K} \sum_{1\leq k\leq K}V^{n(k)}f$ do not converge a.s. for some function $f\in L_2[0, 1).$

AMS Subject Classification (1991): 47A35

Keyword(s): subsequence, ergodic means, ergodic type theorem

Received January 19, 2004, and in revised form August 2, 2005. (Registered under 5897/2009.)

Abstract. Let $A, B$, and $T$ be bounded operators on a separable Hilbert space. Suppose that $A$ and $B$ are normal and that $f$ is a Hölder$(\alpha)$ function with Hölder constant $\lbrack f \rbrack$ defined on the square $D$ containing the spectra of both $A$ and $B$. Then $$\eqalign{\|f(A)T-Tf(B)\| &\leq8\lbrack f \rbrack2^{1-\alpha} \frac{(\sqrt{2}+1)^{\alpha}}{\alpha+1} \|T\|^{1-\alpha} \|AT-TB\|^{\alpha}\times\cr &\hskip30pt \times\big(\log\big(\frac{d\|T\|}{\|AT-TB\|}+1\big)+2\big)^{\alpha+2}, }$$ where $d$ is the side of the square $D$.

AMS Subject Classification (1991): 47B35

Received December 28, 2004. (Registered under 5898/2009.)

Abstract. In this paper, we study spectral properties of class $wF(p,r,q)$ operators for $p+r\leq1$ and $q\geq1$. We show that if $T$ belongs to class $wF(p,r,q)$ operators, then the nonzero points of its point spectrum and joint point spectrum are identical, the nonzero points of its approximate point spectrum and joint approximate point spectrum are identical, and Putnam's theorems hold for class $wF(p,r,q)$ operators.

AMS Subject Classification (1991): 47B20, 47A30

Keyword(s): wF(p, class, r, q), (joint) point spectrum, (joint) approximate point spectrum, Berberian's theorem, Putnam's theorems

Received July 28, and in revised form December 30, 2004. (Registered under 5899/2009.)

Abstract. Let $\phi $ be a bijective continuous map on the algebra of all $n\times n$ matrices, $n\ge2$, preserving commutativity in both directions (no linearity is assumed). Then $\phi $ is a similarity transformation composed with a locally polynomial map, possibly composed with the transposition and the entrywise complex conjugation. The main tool in the proof is the characterization of bijective maps defined on rank one idempotents that preserve orthogonality in both directions. This result, related to some problems in quantum mechanics, is considered also in the infinite-dimensional setting.

AMS Subject Classification (1991): 15A27, 47B49, 51A05

Received May 13, 2005, and in revised form September 16, 2005. (Registered under 5900/2009.)

Abstract. Inhomogeneous Galton--Watson branching mechanisms with immigration are investigated, where the offspring mean tends to its critical value, the offspring variance tends to zero, and the rates of convergences depend on time in both cases. A functional central limit theorem is proved and it is shown that the limit is an inhomogeneous Ornstein--Uhlenbeck type diffusion. The result is applied for simulation of Ornstein--Uhlenbeck type diffusions by Bernoulli trials.

AMS Subject Classification (1991): 60J80, 60J60

Received September 17, 2004, and in revised form March 30, 2005. (Registered under 5901/2009.)