ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. The concept of translations on graphs is introduced. Graphs, where every block is a complete graph, median graphs and the covering graphs of finite distributive lattices graphs are characterized by means of special translations and other mappings.

AMS Subject Classification (1991): 05C12, 06B10

Keyword(s): convexes of graphs, translation, medians, lattices

Received October 21, 1998, and in revised form March 1, 2000. (Registered under 2746/2009.)

Abstract. We study a class of direct systems of rings satisfying a lifting property $(L)$ in order to generalize some properties known in $R[\infty ]$, $R(\infty )$ and $R\langle\infty \rangle $. Moreover, the following theorem is given, generalizing that $R\langle\infty \rangle $ and $R(\infty )$ are stably strong $S$ if $R$ has a finite valuative dimension. If $A=\lim_\to(S_j^{- 1}R[\Lambda_j],f_{kj})$ is a locally finite-dimensionsal domain, $f_{kj}$ are $R$-homomorphisms, and t.d.$[A:R]=\infty $, then $A$ is a stably strong $S$-domain. Finally, we present another characterization of rings satisfying the valuative altitude formula.

AMS Subject Classification (1991): 13C05, 13F05, 13F20

Received April 29, 1999, and in revised form March 2, 2000. (Registered under 2747/2009.)

Abstract. Let ${\msbm W}$ be a proper subvariety of a variety ${\msbm V}$. We say that a functor $F\colon{\cal K}\to{\msbm V}$ is a ${\msbm W}$-relatively full embedding if $F$ is faithful, $\mathop{\rm Im} (Ff)\notin{\msbm W}$ for any ${\cal K}$-morphism $f$, and if $f\colon Fa\to Fb$ is a homomorphism for ${\cal K}$-objects $a$ and $b$ then either $\mathop{\rm Im} (f)\in{\msbm W}$ or $f=Fg$ for some ${\cal K}$-morphism $g\colon a\to b$. A variety of algebras ${\msbm V}$ is called var-relatively universal if there exist a proper subvariety ${\msbm W}$ of ${\msbm V}$ and a ${\msbm W}$-relatively full embedding from the category of all graphs and compatible mappings into ${\msbm V}$. We prove that a variety ${\msbm V}$ of bands is var-relatively universal if and only if ${\msbm V}$ contains the variety of all left semi-normal bands or the variety of all right semi-normal bands.

AMS Subject Classification (1991): 18B15, 20M07, 20M15

Keyword(s): full embedding, lattice of varieties of bands, determinacy

Received July 27, 1999, and in revised form April 4, 2000. (Registered under 2748/2009.)

Abstract. McAlister proved that every locally inverse regular semigroup is a locally isomorphic image of a regular Rees matrix semigroup over an inverse semigroup. In this paper, we show how this result can be generalised to a class of locally adequate abundant semigroups.

AMS Subject Classification (1991): 20M10, 20M17

Received September 28, 1995, and in final form June 5, 2000. (Registered under 2749/2009.)

Abstract. If the dimension of the space spanned by the vectors $\langle f_{1}^{(s)}(x)$, $\ldots $, $f_{n}^{(s)}(x)\rangle $, $s=0,1,\ldots,k$, of $n$ real-valued functions $f_{1},\ldots,f_{n}$ and of their first $k$ derivatives is independent of $x\in I$ (an interval $\subseteq{\msbm R}{}$) and is at most $k$, then the space itself is independent of $x\in I$. This was proved by Curtiss and Moszner assuming the continuity of $f_{1}^{(k)},\ldots,f_{n}^{(k)}$. Their proofs are simplified and extended to operator-valued maps. The extension relies on this generalization of a theorem of Peano: Let $T\colon I\to L(V,W)$ be a differentiable map from a nondegenerate interval $I\subseteq{\msbm R}$ to the space $L(V,W)$ of linear operators from a real finite-dimensional vector space $V$ to another such space $W$. Then $\mathop{\rm range}T(x)$, $x\in I$, is constant if and only if $\mathop{\rm range}T^{\prime }(x)\subseteq\mathop{\rm range}T(x)$, $\dim\mathop{\rm range}T(x)=\mathop{\rm const}$, $x\in I$.

AMS Subject Classification (1991): 15A15, 26A06, 15A04

Received June 23, 1999, and in revised form December 2, 1999. (Registered under 2750/2009.)

Abstract. Let $p>0$. A Borel function $f$, locally integrable in the unit ball $B$, is said to be a $BMO_p(B)$ function if $$||f||_{BMO_p}=\sup_{B(a,r)\subset B}\big(\frac{1}{V(B(a,r))}\int_{B(a,r)}|f(x)-f_{B(a,r)}|^pdV(x)\big)^{1/p}<+\infty,$$ where the supremum is taken over all balls $B(a,r)$ in $B$, and $f_{B(a,r)}$ is the mean value of $f$ over $B(a,r)$. Let ${\cal H}(B)$ denote the set of harmonic functions in open unit ball $B$, $f_{a,r}(x)$ denotes $f(a+rx)$ for arbitrary function $f$. The main result of this paper is to prove the following theorem: Let $u\in{\cal H}(B)$, $p>1$. Then a) $$\eqalign{||u||_{BMO_p}^p=\sup_{{a\in B}\atop{0< r< 1-|a|}}\frac{p(p-1)}{2n(n-2)} \int_B\big(&|u_{a,r}(x)-u_{a,r}(0)|^{p-2} |\nabla u_{a,r}(x)|^2\times\cr &\times(2|x|^{2-n}+(n-2)|x|^2-n)\big)dV_N(x)}$$ for $n\geq3$, and b) $$\eqalign{||u||_{BMO_p}^p=\sup_{{a\in B}\atop{0< r< 1-|a|}}p(p-1)\int_B\big(&|u_{a,r}(x)-u_{a,r}(0)|^{p-2} |\nabla u_{a,r}(x)|^2\times\cr &\times\big(\ln\frac {1}{|x|}-1+|x|\big)\big)dV_N(x)}$$ for $n=2$.

AMS Subject Classification (1991): 31B05, 31C05

Received March 8, 1999. (Registered under 2751/2009.)

Abstract. We consider a system of functional differential equations $x'(t)=F(t,x_t)$ and obtain conditions on a Liapunov functional and a Liapunov function to ensure the stability of the zero solution of functional differential equation with finite delay.

AMS Subject Classification (1991): 34K20

Keyword(s): Uniform asymptotic stability, functional differential equations

Received April 1, 1998, and in revised form December 16, 1999. (Registered under 2752/2009.)

Abstract. It is proved that --- under certain conditions --- solutions $f$ of the functional equation $$ f(x)=h(x,y,f(g_1(x,y)),\ldots,f(g_n(x,y))), (x,y)\in D\subset{{\msbm R}^n}\times{\msbm R}^l $$ having Baire property are continuous, even if $1\le l\le n$. As a tool we introduce new function classes which --- roughly speaking --- interpolate between Baire property and continuity.

AMS Subject Classification (1991): 39B05, 54E52

Received August 12, 1999, and in revised form April 5, 2000. (Registered under 2753/2009.)

Abstract. Let $M$ be a mean on $[a,b]$ and let $\hat M(x,y):=x+y-M(x,y)$ $(x,y\in[a,b])$ be the mean which is complementary to $M$ with respect to the arithmetic mean. A function $f\colon[a,b]\to{\msbm R}$ is called {\it $M$-associate } if it possesses the following property: If $x,y\in[a,b]$ satisfy $M(x,y)=(x+y)/2$ and $f(x )=f\left((x+y)/2\right )$, then $f(y)=f(x)$. We consider the functional equation $$ f(M(x,y))=f(\hat M( x,y)) (x,y\in[a,b]) $$ with and without $f$ being $M$-associate.

AMS Subject Classification (1991): 39B22, 39B12, 26A18

Keyword(s): quasi-arithmetic mean, functional equation

Received May 3, 2000. (Registered under 2754/2009.)

Abstract. We will prove that the maximal Cesàro operator $\sigma_*^\delta$ is bounded from $H^p({{\msbm R}})$ to $L^p({{\msbm R}})$ when $\delta >\delta_p:=1/p-1$, $0< p\leq1$, while $\sigma_*^{\delta_p}$ maps $H^p({{\msbm R}})$ boundedly into {\sl weak}-$L^p({{\msbm R}})$ for $0< p< 1$. The weak type estimate is best possible in the sense that it cannot be strengthened to strong type. The results extend and strengthen those of [7], [11], and [1].

AMS Subject Classification (1991): 42A38, 42A08, 42B30

Keyword(s): Fourier transforms, Cesàro means, Hardy spaces

Received June 23, 1999. (Registered under 2755/2009.)

Abstract. We prove that an operator valued positive definite function defined on an interval of ${\msbm Z}^2$ with the lexicographic order can be extended to a positive definite function on the whole discrete plane.

AMS Subject Classification (1991): 43A35, 47D03

Keyword(s): operator valued positive definite functions, semigroup of operators, lexicographic order

Received April 29, 1999, and in revised form November 8, 1999. (Registered under 2756/2009.)

Abstract. Let ${\eufm t}[\cdot,\cdot ]$ be a densely defined symmetric sesquilinear form in a Hilbert space ${\eufm H}$ with inner product $(\cdot,\cdot )$. Assume that for some $\lambda\in {\msbm R}$ the form ${\eufm t}[\cdot,\cdot ]-\lambda(\cdot,\cdot )$ induces a Kreĭn space structure on $\mathop{\rm dom}{\eufm t}$, which can be continuously embedded in ${\eufm H}$. Then there exists a unique selfadjoint operator $T_{\eufm t}$ in ${\eufm H}$ such that $\mathop{\rm dom}T_{\eufm t}\subset\mathop{\rm dom}{\eufm t}$ and ${\eufm t}[f,g]=(T_{\eufm t}f,g)$, $f \in\mathop{\rm dom}T_{\eufm t}$, $g \in\mathop{\rm dom}{\eufm t}$. This generalizes the first representation theorem in T. Kato [Kato] to a non-semibounded situation. Based on the theory of definitizable operators in Kreĭn spaces an analog of the second representation theorem in [Kato] will be given. These results provide an approach to generalized Friedrichs extensions for a class of non-semibounded symmetric operators with defect numbers $(1,1)$, which is analogous to the classical theory in the semibounded case.

AMS Subject Classification (1991): 46C20, 47A67, 47B50; 47B25

Keyword(s): Sesquilinear form, representation theorem, Kreĭn space, singular critical point, generalized Friedrichs extension

Received May 3, 1999. (Registered under 2757/2009.)

Abstract. We derive a formula of a weight $v$ in terms of a given weight $w$ such that the estimate $$ \int_{\msbm D}| f(z)| ^pw(z) dm(z) \sim | f(0)| ^p + \int_{\msbm D}| f'(z)| ^pv(z) dm(z) $$ is valid for all analytic functions $f$ on the unit disc.

AMS Subject Classification (1991): 46E15, 30E99

Received February 15, 1999, and in revised form March 3, 2000. (Registered under 2758/2009.)

Abstract. Continuous and compact imbeddings of weighted Sobolev spaces $W^{1,p}(\Omega; v)$ and $W_0^{1,p}(\Omega; v)$ into the weighted Lebesgue space $L^q (\Omega; w)$, where $1\le q< p< \infty $, have been considered, where the weights $v$ and $w$ are some functions of the distance measured either from the boundary $\partial\Omega $ of $\Omega $ or from a point $x_0 \in\partial \Omega $ and $\Omega $ is a bounded domain of ${{\msbm R}^N}$ in the class ${{\cal C}^{0,1}}$, ${{\cal K}(x_0), {\cal K}^{0,1}(x_0)}$.

AMS Subject Classification (1991): 46E35

Received October 26, 1998, and in final form September 15, 1999. (Registered under 2759/2009.)

Abstract. This paper concerns strongly irreducible decomposition and irreducible decomposition for Cowen--Douglas operators and operator weighted shifts. We characterize strong irreducibility of an operator weighted shift by the Jacobson radical of its commutant. Moreover, we show that every Cowen--Douglas operator and operator weighted shift has uniquely finite irreducible decomposition under unitary equivalence.

AMS Subject Classification (1991): 46H30, 47A10, 47A55, 47A58

Received November 18, 1998, and in revised form July 6, 1999. (Registered under 2760/2009.)

Abstract. Let $\lambda_1,\ldots,\lambda_n$ be elements of the essential approximate point spectrum of a bounded Banach space operator. Then there are corresponding approximate eigenvectors $x_1,\ldots,x_n$ such that the norm on the subspace generated by them is almost symmetric. The result can be used in the Scott Brown technique for Banach space operators. Another application is for the local behaviour of operators.

AMS Subject Classification (1991): 47A05, 47A10, 47A15

Received July 1, 1999, and in revised form December 13, 1999. (Registered under 2761/2009.)

Abstract. The main purpose of this paper is to present a reducibility result based on a recent theorem of Turovskii on semi-groups of compact quasinilpotent operators. More precisely, we prove that every non-zero triangularizable family of compact operators has a hyperinvariant subspace, and then we present several sufficient conditions for simultaneous triangularization of a family of compact operators together with its commutant. We also give a different proof of Shulman's theorem. The finite-dimensional version of the results is also mentioned and emphasized.

AMS Subject Classification (1991): 47A15, 47D03, 20M20

Keyword(s): Volterra semigroup (algebra), hyperinvariant subspace, Commutant, Triangularization

Received August 10, 1999. (Registered under 2762/2009.)

Abstract. As shown by Berger, Coburn and Lebow [1] and recently rediscovered by Douglas and Foiaş [3] every c.n.u. bi-isometry is unitarily equivalent with a certain isometric pair on $H^2({\msbm T},{\cal E})$ ($\cal E$ is a Hilbert space) defined in terms of two operators on ${\cal E}$, $U$ unitary and $P$ orthogonal projection. It is our aim in this paper to characterize the structure of a double commuting c.n.u. bi-isometry related to the Wold-Słociński decomposition [10] in terms of a representative $\{U,P\} $ of its complete unitary invariant. Some results concerning the minimal unitary extension are also given.

AMS Subject Classification (1991): 47A45

Received October 18, 1999. (Registered under 2763/2009.)

Abstract. We have two typical examples of semi-hyponormal but not hyponormal operators. In this paper, we show that these examples have the following property: Re $\sigma(T) = \sigma(\mathop{\rm Re }T)$.

AMS Subject Classification (1991): 47B20

Received October 24, 1999, and in revised form April 19, 2000. (Registered under 2764/2009.)

Abstract. An operator $X\colon{\cal H}_1 \to{\cal H}_2$ is said to be a generalized Toeplitz operator with respect to given contractions $T_1$ and $T_2$ if $X=T_2XT_1^*$. The purpose of this line of research, started by Douglas, Sz.-Nagy and Foiaş, and Pták and Vrbová, is to study which properties of classical Toeplitz operators depend on their characteristic relation. Following this spirit, we give some clarifying examples and a new characterization of analytic Toeplitz operators that complement the work done by Pták and Vrbová, as well as some spectral properties of generalized Toeplitz operators that complement the work done by Sz.-Nagy and Foiaş. As a by-product we prove that the spectrum of a function $\phi\in H^\infty $ equals the approximate point spectrum of its Toeplitz operator.

AMS Subject Classification (1991): 47B35

Keyword(s): Toeplitz operators, spectral properties, minimal isometric dilation

Received December 21, 1998, and in revised form November 18, 1999. (Registered under 2765/2009.)

Abstract. If $\varphi $ is an analytic map of the unit disk $D$ into itself, the composition operator $C_{\varphi }$ on the Hardy space $H^2(D)$ is defined by $C_{\varphi}(f) = f\circ\varphi$. For a certain class of composition operators with multivalent symbol $\varphi$, we identify a subspace of $H^2(D)$ on which $C^*_{\varphi}$ behaves like a weighted shift. We reproduce the description of the spectrum found in [Kam75] and show for this class of composition operators that the interior of the spectrum is a disk of eigenvalues of $C^*_{\varphi}$ of infinite multiplicity.

AMS Subject Classification (1991): 47B38

Received February 9, 1999. (Registered under 2766/2009.)

Abstract. The concept of an elementary operator between two algebras was recently introduced and this paper continues and extends the study of this concept. Elementary operators on some function algebras are computed. Jordan elementary operators are introduced and, in particular, their form on standard operator algebras is described.

AMS Subject Classification (1991): 47B47, 46E25, 16W99

Received December 10, 1999. (Registered under 2767/2009.)

Abstract. Given a complex Hilbert space $X$ and the von Nuemann algebra ${\cal L}(X)$, we study the Riemannian geometry of the manifold ${\cal P}(X)$ consisting of all minimal projections in ${\cal L}(X)$. To do it we take the Jordan--Banach triple approach (briefly, the JB$^*$-triple approach) because this setting provides a unifying framework for many other situations and simplifies the study previously made by other authors. We then apply this method to study the differential geometry of the manifold of minimal partial isometries in ${\cal L}(H, K)$, the space of bounded linear operators between the complex Hilbert spaces $H$ and $K$ with $\dim H \leq\dim K$.

AMS Subject Classification (1991): 17C36, 53C22

Keyword(s): Partial isometries, JB*-triples, Affine connections, Geodesics, Riemannian distance

Receved July 5, 1999, and in revised form November 12, 1999. (Registered under 2768/2009.)

Abstract. Voiculescu's asymptotic freeness result for random matrices is improved to the sense of almost everywhere convergence. The asymptotic freeness almost everywhere is first shown for standard unitary matrices based on the computation of multiple moments of their entries, and then it is shown for rather general unitarily invariant selfadjoint random matrices (in particular, standard selfadjoint Gaussian matrices) by applying the first result to the unitary parts of their diagonalization. Bi-unitarily invariant non-selfadjoint random matrices are also treated via polar decomposition.

AMS Subject Classification (1991): 15A52, 62E20, 60F99

Keyword(s): random matrices, free probability, asymptotic freeness

Received January 14, 2000. (Registered under 2769/2009.)