ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. We give a new proof of a theorem of József Dénes: If $L_1$ and $L_2$ are distinct latin squares of order $n \ge2$, $n \notin\{4,6\} $, that satisfy the quadrangle criterion, then $L_1$ and $L_2$ differ in at least $2n$ entries.

AMS Subject Classification (1991): 20D60, 05B15

Keyword(s): multiplication table, quadrangle criterion, Hamming distance

Received December 12, 2002, and in revised form March 19, 2003. (Registered under 5794/2009.)

Abstract. In this paper we investigate generalized higher-order convex functions. Our main results extend the classical Hermite--Hadamard inequality for this setting.

AMS Subject Classification (1991): 26A51, 26B25

Received August 19, 2003. (Registered under 5795/2009.)

Abstract. We prove an inequality for sums, which generalizes both Landau's sharpening of Carlson's inequality and the corresponding complementary result by Levin and Stečkin. The inequality is optimal, in the sense that necessary and sufficient conditions on the parameters for which the inequality holds are given. In some cases, sharp constants are obtained, also in situations not covered by the classical results.

AMS Subject Classification (1991): 26D15

Keyword(s): Inequalities, Landau inequality, Levin--Stečkin inequality

Received June 26, 2003. (Registered under 5796/2009.)

Abstract. Let $w(x):=\exp(-Q(x))$ be a non-negative weight function on the closed set $\Sigma\subset {\msbm R}$. We will introduce some conditions, each of which guarantees that the support of the equilibrium measure associated with $w$ is an interval. For example: if $\exp(Q(x))$ is convex, then the support of the equilibrium measure is an interval. These type of conditions are very useful when determining the equilibrium measure. An integral representation formula for the equilibrium measure is also given.

AMS Subject Classification (1991): 31A15

Received September 22, 2003, and in revised form November 28, 2003. (Registered under 5797/2009.)

Abstract. The functions $f_{1},\ldots,f_{n}$ satisfy a $k$th-order explicit homogeneous linear differential equation with continuous coefficients on a nondegenerate interval $I$ if and only if they are $k$ times continuously differentiable and $$\mathop{\rm rank}{\bf M}_{k-1}( f_{1},\ldots,f_{n};x) =\mathop{\rm rank}{\bf M}_{k}( f_{1},\ldots,f_{n};x) =\mathop{\rm const}, x\in I,$$ where ${\bf M}_{m}( f_{1},\ldots,f_{n};x)$ is the $m$th-order Wronskian matrix of $f_{1},\ldots,f_{n}$, whose rows are the successive derivatives of $f_{1},\ldots,f_{n}$. The equation is unique if its vector of coefficients is a linear combination of the $(k-1) $-jets $\langle f_{i}(x),\ldots,f_{i}^{(k-1)}(x)\rangle $ of the $f_{i}$ for every $x\in I$. The coefficients of this unique equation are infinitely differentiable or analytic if $f_{1},\ldots,f_{n}$ are such. If $f_{1},\ldots,f_{n}$ are linearly independent, then the equation is written explicitly. The results are extended to first-order systems.

AMS Subject Classification (1991): 34A30, 15A15

Keyword(s): explicit homogeneous linear differential equations with continuous coefficients, Wronskian matrices

Received December 30, 2002, and in revised form June 25, 2003. (Registered under 5798/2009.)

Abstract. In the present paper, the existence, uniqueness and continuous dependence upon the data of the strong solution of a mixed problem for a singular second order hyperbolic equation with an integral condition is proved. The proof is essentially based on an a priori estimate and on the density of the range of the operator generated by the considered problem. In spite of the apparent simplicity of the problem, the solution requires a delicate set of techniques. The investigation is accomplished in a traditional functional analysis language and techniques.

AMS Subject Classification (1991): 35L20, 35L67

Keyword(s): Singular hyperbolic equation, Integral condition, Strong solution

Received January 3, 2001, and in revised form March 6, 2003. (Registered under 5799/2009.)

Abstract. The well-known Paley type inequality $$\Big(\sum_{i=0}^\infty2^{i(2-2/p)} |{\hat f}(2^i)|^2 \Big)^{1/2} \leq C_p\|f\|_{H^p}\qquad(0< p\leq1)$$ is generalized for one- and more-parameter Ciesielski--Fourier coefficients and for Hardy spaces.

AMS Subject Classification (1991): 41A15, 42B05, 42C10, 42B30

Keyword(s): Paley type inequalities, multipliers, Walsh system, spline functions, Ciesielski system, Hardy spaces, atomic decomposition

Received September 5, 2003. (Registered under 5800/2009.)

Abstract. We give necessary and sufficient conditions in order that the sum of a double cosine series with nonnegative coefficients belong to the two-dimensional Zygmund classes $\Lambda_*(2)$ or $\lambda_*(2)$, respectively. Our theorem is the extension of the corresponding theorem by Boas from single to double cosine series.

AMS Subject Classification (1991): 42B05, 42A16

Keyword(s): cosine series, \Lambda_*, \lambda_*, Zygmund classesand, summation by parts

Received February 6, 2003, and in revised form April 11, 2003. (Registered under 5801/2009.)

Abstract. We give necessary and sufficient coefficient conditions under which the sum of a double sine or cosine-sine series with nonnegative coefficients belong to the two-dimensional Zygmund classes $\Lambda_*(2)$ or $\lambda_*(2)$, respectively. Our theorems are the extensions of the corresponding theorems by Boas and by Németh from single to double series.

AMS Subject Classification (1991): 42B05, 42A16

Keyword(s): cosine series, sine series, \Lambda_*, \lambda_*, Zygmund classesand, summation by parts

Received June 11, 2003. (Registered under 5802/2009.)

Abstract. The conventional monotonicity hypothesis on the coefficients of an orthogonal series given in two theorems of K. Tandori will be weakened to locally almost monotonicity assumption.

AMS Subject Classification (1991): 42C15, 42C05

Received February 13, 2003. (Registered under 5803/2009.)

Abstract. In this paper we put together some applications given in [7] and [8]. First we deal with the notion of a basis of $\alpha $, $\beta $ type and with the expansion of a function in a power series. Then we deal with the finite section method for operators mapping $s_{\alpha }$ into itself.

AMS Subject Classification (1991): 46A15

Keyword(s): Infinite linear system, operator of first-difference, unital Banach algebra, BK space

Received November 1, 2002, and in final form March 17, 2003. (Registered under 5804/2009.)

Abstract. On the Hardy space over the torus $H^2(\Gamma ^2)$, the Toeplitz operators $T_{z}$ and $T_{w}$ are unilateral shifts of infinite multiplicity. Subspaces $N\subset H^2(\Gamma ^2)$ invariant under $T^*_{z}$ and $T^*_{w}$ are said to be backward shift invariant. This paper studies the compression of the pair $(T_{z}, T_{w})$ (denoted by $(S_{z}, S_{w})$) to $N$. Its focus lies on the case when $S_z$ is a strict contraction. Much information about $N^{\perp }$ can be deduced in this case.

AMS Subject Classification (1991): 46E20, 47A20, 47A13

Received December 30, 2002, and in final form April 3, 2003. (Registered under 5805/2009.)

Abstract. Let $(S,\Sigma,\lambda )$ be the usual Lebesgue measure space of the unit interval $S, X$ a real Banach space and $\mu $ a vector measure on $\Sigma $ into $X$ absolutely continuous with respect to $\lambda $. If the associated integration map $T$ of $\mu $ extends to and is bounded on $L^p(\lambda )$ for some finite $p$ then every weakly null sequence in the (closed convex hull of the) range of $\mu $ admits a subsequence in $\ell ^2_{\rm weak}(X)$. This is only a sufficient condition; also, in general we cannot do better than $\ell ^2_{\rm weak}(X)$. Further, in every infinite dimensional Banach space $X$ the measures $\mu $ whose $T$'s do not extend in this manner form a residual set in the Banach space $cca(\lambda )$ of all measures from $\Sigma $ into $X$ with relatively norm compact ranges, absolutely continuous with respect to $\lambda $, under the semi variation norm.

AMS Subject Classification (1991): 46G10, 28B45

Received June 26, 2003, and in revised form August 22, 2003. (Registered under 5806/2009.)

Abstract. Let $A,B$ and $C$ be unital $C^{\ast }$-algebras with $B$ injective. Let $C$ be a subalgebra of $A$ and $B$ with $I_{C}=I_{A}$ and $I_{C}=I_{B},$ let $M$ be a complex subspace of $A$ with $c_{1}Mc_{2}\subseteq M$, for all $c_{1}, c_{2}\in C$, and let $L\colon M\to B$ be a $w_{2}$ completely bounded $C$-bihomomorphism. Then there exists a $C$-bihomomorphism extension $\widetilde{L}\colon A\to B$ of $L$ with $ \|\widetilde{L} \|_{w_{2}cb}= \|L \|_{w_{2}cb}.$ We also prove an extension theorem for a self-adjoint $w_{\rho }$ completely bounded $C$-bihomomorphism on a subspace of a unital $C^{\ast }$-algebra with $0< \rho\leq 2$.

AMS Subject Classification (1991): 46L05

Keyword(s): C, -bihomomorphism, w_{\rho }, contraction, w_{\rho }, completely bounded map

Received November 1, 2002. (Registered under 5807/2009.)

Abstract. Let ${\cal A}$ be a unital, norm-closed subalgebra of the bounded operators ${\msbm B}({\cal H})$ on a Hilbert space ${\cal H}$ and $T$ a normal left ${\cal A}^*$-, right ${\cal A}$-module completely bounded map of ${\msbm B}({\cal H})$. For the numerical radius norm $w(\cdot )$ on ${\msbm B}({\cal H})$, we let $\|T\|_w = \sup\{w(T(x)) | w(x) \le1 \} $ and $\|T\|_{wcb}= \sup_n \|T \otimes\mathop{\rm id}_n \|_w$. It is shown that there exist $t=(t_{ij})\in{\msbm B}(\ell ^2(I))$ and elements $v_i$ ($i \in I$) of the commutant of ${\cal A}$ such that $\|t\|=\|T\|_{wcb}$, $\sum_{i\in I}v_i^*v_i \le1$, and $T(x) = \sum_{i,j\in I}v_i^*t_{ij}xv_j$ $(x\in{\msbm B}({\cal H}))$. As an application, we generalize Ando--Okubo's theorem for Schur multipliers on ${\msbm B}({\cal H})$.

AMS Subject Classification (1991): 46L05, 46L07, 47C15

Keyword(s): Completely bounded, Completely positive, Schur multipliers, Numerical radius norm, Operator systems

Received May 12, 2003, and in revised form October 29, 2003. (Registered under 5808/2009.)

Abstract. The notion of bundle convergence for single (ordinary) sequences in von Neumann algebras and their $L_2$-spaces was introduced by Hensz, Jajte and Paszkiewicz in 1996. We adopted this notion for double sequences in 2001. In the present paper, we prove a new two-parameter SLLN for double sequences of orthogonal vectors in an $L_2$-space, and this SLLN is an intermediate one between those two Strong Laws of Large Numbers (in abbreviation: SLLN) proved in [4, Theorems 1 and 2]. One of our tools is a more precise variant of the Rademacher--Menshov inequality in noncommutative setting, which may be useful in other cases.

AMS Subject Classification (1991): 46L10, 46L53, 60B12

Keyword(s): von Neumann algebra, Gelfand-Naimark-Segal representation theorem, bundle convergence, orthogonal vectors, Rademacher-Menshov inequality, L_2, SLLN in noncommutative-spaces

Received September 13, 2002, and in revised form December 30, 2003. (Registered under 5809/2009.)

Abstract. Let ${\cal A}$ be a complex normed algebra. For $A,B\in{\cal A}$, define a basic elementary operator $M_{A,B}\colon{\cal A} \rightarrow{\cal A}$ by $M_{A,B}(X)=AXB$. Given a standard operator algebra ${\cal A}$ acting on a complex normed space and $A,B\in{\cal A}$ we have: (i) The lower estimate $ \|M_{A,B}+M_{B,A} \|\geq2(\sqrt{2}-1) \|A \|\|B \|$ holds. (ii) The lower estimate $ \|M_{A,B}+M_{B,A} \|\geq\|A \|\|B \|$ holds if $$\inf_{\lambda\in C} \|A+\lambda B \|= \|A \|\hbox{ or } \inf_{\lambda\in C} \|B+\lambda A \|= \|B \|.$$ (iii) The equality $ \|M_{A,B}+M_{B,A} \|=2 \|A \|\|B \|$ holds if $$ \|A+\lambda B \|= \|A \|+ \|B \|\hbox{ for some unit scalar }\lambda.$$ These results extend analogous estimates established earlier for standard operator subalgebras of Hilbert space operators.

AMS Subject Classification (1991): 46L35, 47L35, 47B47

Keyword(s): Lower estimate, standard operator algebra, elementary operator

Received August 21, 2003, and in revised form November 11, 2003. (Registered under 5810/2009.)

Abstract. It is shown that the complex tangent space $R_a$ at a point $a$ on the surface of the unit ball $A_1$ in a complex Banach space $A$ coincides with the complex linear span $$\mathop{\rm lin}_{{\msbm C}}(\{{\rm i} a\}^{\rlap{\sqcap}\sqcup} \cap\{a\}^{\rlap{\sqcap}\sqcup} \cap A_1)$$ of the set $\{{\rm i} a\}^{\rlap{\sqcap}\sqcup} \cap\{a\}^{\rlap{\sqcap}\sqcup} \cap A_1$, where, for a subset $L$ of $A$, \[L^{\rlap{\sqcap}\sqcup} = \{a \in A: \|a \pm b\| = \max\{\|a\|,\|b\|\}, \forall b \in L\}\] is the M-orthogonal complement of $L$. It is also shown that if $B$ is a holomorphically rigid closed subspace of $A$ then $B^{\rlap{\sqcap}\sqcup}$ is equal to $\{0\}$. In the special case in which $A$ is a JBW$^*$-triple and $B$ is a weak$^*$-closed subtriple of $A$, it is shown that the M-orthogonal complement $B^{\rlap{\sqcap}\sqcup}$ of $B$ coincides with the algebraic annihilator $B^{\perp}$ of $B$, that the complex tangent space $R_{L_B}(A)$ at the set $L_B$ of elements of $B$ of unit norm is weak$^*$-closed and also coincides with $B^{\rlap{\sqcap}\sqcup}$, that a second tangent space $T^n_{L_B}(A)$ at $L_B$ is weak*-closed and coincides with the algebraic kernel $\mathop{\rm Ker}(B)$ of $B$, and that $B$ is holomorphically rigid in $A$ if and only if $B^{\rlap{\sqcap}\sqcup}$ is equal to $\{0\}$.

AMS Subject Classification (1991): 46L70; 17C65, 46G20

Keyword(s): M-structure, tangent spaces, holomorphic rigidity

Received April 8, 2003, and in revised form October 14, 2003. (Registered under 5811/2009.)

Abstract. We study the semi-Browder spectra and other spectra originating from Fredholm theory in the particular case that a bounded operator $T$ on a Banach space, or its dual $T^\star $, admits the single valued extension property.

AMS Subject Classification (1991): 47A10, 47A11, 47A53, 47A55

Keyword(s): Single valued extension property, Fredholm theory

Received January 21, 2003. (Registered under 5812/2009.)

Abstract. Let $T = U|T|$ be the polar decomposition of a bounded linear operator $T$ on a complex Hilbert space ${\cal H}$ and $T(s,t)=|T|^sU|T|^t$ for $ 0 < s, t \leq1$. $T$ is called a class $A(s,t)$ operator if $ |T(s,t)|^{2t\over s+t} \geq |T|^{2t} $, which is a further generalization of $p$-hyponormal or $\log $-hyponormal operator. We shall show that if $T$ is a class $A(s,t)$ operator for $ 0 < s, t\leq1$, then (i) $ \sigma(T(s,t)) = \{r^{s+t}e^{i\theta } : re^{i\theta } \in\sigma (T)\} $, (ii) for each non-zero isolated point $\lambda =re^{i\theta }$ of $\sigma(T)$, the Riesz idempotent $E$ for $T$ with respect to $\lambda $ is self-adjoint and coincides with the Riesz idempotent $E(s,t)$ for $T(s,t)$ with respect to $\lambda_{s+t}=r^{s+t}e^{i\theta }$.

AMS Subject Classification (1991): 47A10, 47B20

Keyword(s): A, classoperator, A(s, class, operator, t), Riesz idempotent

Received September 30, 2003. (Registered under 5813/2009.)

Abstract. A generalization of Turovskii's results for Volterra semigroups to locally convex spaces and some consequences are given.

AMS Subject Classification (1991): 47A15, 47B99, 46A32

Keyword(s): Locally convex space, compact operator, quasinilpotent operator, operator semigroup, invariant subspace, triangularization

Received March 12, 2003, and in final form January 26, 2004. (Registered under 5814/2009.)

Abstract. Completing research made in [K2], we characterize vectors of a Hilbert space, which have a canonical, inner--outer-type factorization in relation to a given operator $T$. Special emphasis is put on the case, when $T$ is an operator admitting an $H^{\infty }$-functional calculus, and, in particular, when $T$ is quasisimilar to the unilateral shift.

AMS Subject Classification (1991): 47A16, 47A45, 47A60, 47B35

Keyword(s): Paley type inequalities, multipliers, Walsh system, spline functions, Ciesielski system, Hardy spaces, atomic decomposition

Received October 10, 2003. (Registered under 5815/2009.)

Abstract. In what follows, an operator means a bounded linear operator on a Hilbert space {\it H}. We shall investigate several basic properties of the {\it generalized Kantorovich constant } and its applications to related results. Firstly we shall show that Kantorovich type inequality for $1 > p>0$ and Kantorovich type one for $ p< 0$ are both obtained by Kantorovich type one for $p>1$. Secondly we shall show that these three Kantorovich type inequalities are mutually equivalent.

AMS Subject Classification (1991): 47A63

Keyword(s): Generalized Kantorovich constant, Kantorovich type inequality, Specht ratio

Received November 14, 2002, and in revised form December 12, 2002. (Registered under 5816/2009.)

Abstract. Let $T$ be a bounded linear operator on a separable infinite-dimensional Banach space $X$, and let $N(T)$ denote the nullspace of $T$. We say that $T$ is an abstract backward shift of multiplicity $m$, where $1 \leq m < \infty $, if (1) $\dim N(T^n)/N(T^{n-1}) =m$ for all $n \geq1$ and (2) $\cup_{n=1}^\infty N(T^n)$ is dense in $X$. We characterize the commutant of such an operator, and use our result to determine sufficient conditions for the operator to be irreducible.

AMS Subject Classification (1991): 47A99, 47B37

Keyword(s): backward shift, commutant

Received February 13, 2003, and in revised form 26, 2003. (Registered under 5817/2009.)

Abstract. In this note we characterize compact and weakly compact multiplication operators $M_{\pi }$ on the weighted locally convex spaces $CV_{0}(X,E)$ of vector-valued continuous functions induced by the operator-valued mappings $\pi\colon X \rightarrow B(E)$.

AMS Subject Classification (1991): 47B07, 47B33, 47B37, 47B38; 47A56, 46E10, 46E40

Keyword(s): Compact operators, Weakly compact operators, Multiplication operators, Weighted locally convex spaces, Operator-valued mappings

Received August 30, 2002. (Registered under 5818/2009.)

Abstract. In this note we give several examples related to products of Toeplitz operators on the Bergman space. If $f$, $g$, and $h$ are symbols, we say that $T_fT_g=T_h$ in a non-trivial way if neither $\overline f$ nor $g$ is holomorphic. It is known that such triples exist. We give a method to construct many such examples. We show that it is also possible to have $T_fT_g=T_{fg}$ or $T_fT_g=I$ in a non-trivial way. We also have some positive results on products of Toeplitz operators.

AMS Subject Classification (1991): 47B35

Received April 25, 2003, and in revised form September 24, 2003. (Registered under 5819/2009.)

Abstract. The problem of localization of spectral singularities of dissipative operators in terms of the asymptotic of the corresponding exponential function is studied. We give a solution to this problem for the singularities of higher orders in the frame of perturbation theory.

AMS Subject Classification (1991): 47B44, 47D06

Received March 12, 2003. (Registered under 5820/2009.)

Abstract. In the physical papers [1]--[4] there are some interesting propositions about operators acting in Hilbert spaces. For example in [3] there were considered all bijective mappings on ${\cal B}_s(H)$, which preserve the order in both directions. These mappings have been characterized with the help of elementary operators of length 1. The above mentioned papers have motivated the investigation in the present paper.

AMS Subject Classification (1991): 47B47, 47B15

Received June 26, 2003, and in revised form September 24, 2003. (Registered under 5821/2009.)

Abstract. Chen [3] established a sharp inequality for the warping function of a warped product submanifold in a Riemannian space form in terms of the squared mean curvature. Later, in [5], he studied warped product submanifolds in complex hyperbolic spaces. In the present paper, we establish an inequality between the warping function $f$ (intrinsic structure) and the squared mean curvature $\|H\|^2$ and the holomorphic sectional curvature $c$ (extrinsic structures) for warped product submanifolds $M_1\times_fM_2$ with $J{\cal D}_1\perp{\cal D}_2$ (in particular, $CR$-warped product submanifolds and $CR$-Riemannian products) in any complex space form $\widetilde M(c)$. Examples of such submanifolds which satisfy the equality case are given.

AMS Subject Classification (1991): 53C40, 53C42, 53B25

Keyword(s): Warped products, CR, -warped products, CR, -products, warping function

Received May 30, 2003, and in revised form September 22, 2003. (Registered under 5822/2009.)