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XX conference

About multipliers of matrix of operators connected with interpolation spaces

Kabanko M.V., Myagchenkova E.L.

Kursk state university, Faculty of Physics, Mathematics, IT, Dept. Mathematical Analysis and Applied Mathematics, Russia, 305 000, Kursk, Radischeva st., 33, Phon. +7(4712)56-80-61 E-mail: kabankom@mail.ru

1 pp. (accepted)

It's known that multipliers of the matrix of the linear bounded operators in Hilbert space not always are bounded operators on algebra $B(H)$. The most famous example such multiplier is operator triangular truncation. The operator is product of the matrix of the operator and characteristic function of indexes $\chi_{\Delta}(i,j)$ where $\Delta=\{(i,j)\in \mathbb{N}|j\leq i\}$. For norm this operator estimates $\frac{1}{c}\ln(1+n)\leq \|\chi_{\Delta_n}\|\leq c\ln(1+n)$, where $\Delta=\{(i,j)\in \mathbb{N}|j\leq i\leq n\}$ is true. This estimates was obtained V.I. Macaev (see f.e. \cite{Dav}).

Let $\overline{H}=\{H_0,H_1\}$ be Hilbert couple, where $H_0=l_2(2^{-n}G_n)$, $H_1=l_2(2^{n}G_n)$ and $G_n=\mathbb{C}$ for all $n\in \mathbb{Z}$. Denote $B(\overline{H})$ algebra of linear bounded operators that is acting in couple $\overline{H}$. It's proved (see f.e. \cite{K}) if operator $A\in B(\overline{H})$ then elements of matrix of the operator satisfies weighted conditions, more precisely, $|a_{ij}|\leq \|A\|_{B(\overline{H})}2^{-|i-j|}$. So elements that is standing at diagonals of the matrix is completely bounded.

Now let's study representation of algebra $B(\overline{H})$ in interpolation spaces between $H_0$ and $H_1$, for example in $l_2(G_n)$, which is denoted $(l_2(G_n),\varphi)$ (see \cite{KO}). Obviously that image $\varphi(B(\overline{H}))$ be subalgebra in $B(l_2(G_n))$.

Let $M=(m_{ij})_{i,j=-\infty}^{\infty}$ be any matrix then denote $M\circ A$ -- Hadamard-Schur product. Operator $M\circ A$ has matrix $(m_{ij}a_{ij})_{i,j=-\infty}^{\infty}$ with respect to operator $A$ .

{\bf Theorem.} If $A\in \varphi(B(\overline{H}))$ then operator $\chi_{\Delta}(i,j)\circ A$ is bounded in space $l_2(G_n)$.

This means that operator $\chi_{\Delta}(i,j):\varphi(B(\overline{H}))\longrightarrow l_2(G_n)$ is Schur multiplier. Moreover, it is easy to prove, that if $M\in l_{\infty}(\mathbb{Z}^2)$, $A\in \varphi(B(\overline{H}))$ then $M\circ A$ is Schur multiplier too. So in algebra $B(l_2(G_n))$ exists natural subalgebra $\varphi(B(\overline{H}))$ for which all elements of algebra $l_{\infty}(\mathbb{Z}^2)$ is Schur multiplier.

\begin{thebibliography}{100} \bibitem{Dav} \textit{Davidson K.} Nest algebras. Tringular forms for operator algebras on Hilbert space// \textit{Pitman Res. Notes Math. Ser.} \textbf{191}, Longman Sci. and Tech., Harlow, 1988. \bibitem{K} \textit{Kabanko M.V.} Algebra of operators acting in Hilbert couple// \textit{Proceedings of mathematical faculty VSU} \textbf{6}, Voronezh, 2001. P. 54-61. \bibitem{KO} \textit{Kabanko M.V., Ovchinnikov V.I.} On some representations of algebra of operators in Hilbert couple// \textit{Proceedings of mathematical faculty VSU} \textbf{5}, Voronezh, 2001. P. 32-40. \end{thebibliography}



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