• Title/Summary/Keyword: the Hilbert-Schmidt

Search Result 17, Processing Time 0.018 seconds

HILBERT-SCHMIDT INTERPOLATION FOR OPERATORS IN TRIDIAGONAL ALGEBRAS

  • Kang, Joo-Ho;Kim, Ki-Sook
    • Journal of applied mathematics & informatics
    • /
    • v.10 no.1_2
    • /
    • pp.227-233
    • /
    • 2002
  • Given operators X and Y acting on a Hilbert space H, an interpolating operator is a bounded operator A such that AX = Y. An interpolating operator for n-operators satisfies the equation AX$\sub$i/=Y$\sub$i/, for i=1,2, ‥‥, R. In this article, we investigate Hilbert-Schmidt interpolation for operators in tridiagonal algebras.

AN EXTENSION OF THE FUGLEDE-PUTNAM THEOREM TO p-QUASITHYPONORMAL OPERATORS

  • Lee, Mi-Young;Lee, Sang-Hun
    • Bulletin of the Korean Mathematical Society
    • /
    • v.35 no.2
    • /
    • pp.319-324
    • /
    • 1998
  • The equation AX = BX implies $A^*X\;=\;B^X$ when A and B are normal (Fuglede-Putnam theorem). In this paper, the hypotheses on A and B can be relaxed by usin a Hilbert-Schmidt operator X: Let A be p-quasihyponormal and let $B^*$ be invertible p-quasihyponormal such that AX = XB for a Hilbert-Schmidt operator X and $|||A^*|^{1-p}||{\cdot}|||B^{-1}|^{1-p}||\;{\leq}\;1$.Then $A^*X\;=\;XB^*$.

  • PDF

INTERPOLATION FOR HILBERT-SCHMIDT OPERATOR AND APPLICATION TO OPERATOR CORONA THEOREM

  • Kang, Joo-Ho;Ha, Dae-Yeon;Baik, Hyoung-Gu
    • Journal of applied mathematics & informatics
    • /
    • v.9 no.1
    • /
    • pp.341-347
    • /
    • 2002
  • Given operators X and Y acting on a Hilbert space H, an interpolating operator is a bounded operator A such that AX = Y. An interpolating operator for n-operators satisfies the equation $AX_i = Y_i$, for i = 1,2…, n. In this paper, we investigate Hilbert-Schmidt interpolation problems in tridiagonal algebra by connecting the classical corona theorem.

HILBERT-SCHMIDT INTERPOLATION ON Ax = y IN A TRIDIAGONAL ALGEBRA ALGL

  • Jo, Young-Soo;Kang, Joo-Ho
    • The Pure and Applied Mathematics
    • /
    • v.11 no.2
    • /
    • pp.167-173
    • /
    • 2004
  • Given vectors x and y in a separable Hilbert space $\cal H$, an interpolating operator is a bounded operator A such that Ax = y. In this article, we investigate Hilbert-Schmidt interpolation problems for vectors in a tridiagonal algebra. We show the following: Let $\cal L$ be a subspace lattice acting on a separable complex Hilbert space $\cal H$ and let x = ($x_{i}$) and y = ($y_{i}$) be vectors in $\cal H$. Then the following are equivalent; (1) There exists a Hilbert-Schmidt operator A = ($a_{ij}$ in Alg$\cal L$ such that Ax = y. (2) There is a bounded sequence {$a_n$ in C such that ${\sum^{\infty}}_{n=1}\mid\alpha_n\mid^2 < \infty$ and $y_1 = \alpha_1x_1 + \alpha_2x_2$ ... $y_{2k} =\alpha_{4k-1}x_{2k}$ $y_{2k=1} = \alpha_{4kx2k} + \alpha_{4k+1}x_{2k+1} + \alpha_{4k+1}x_{2k+2}$ for K $\epsilon$ N.

  • PDF

HILBERT-SCHMIDT INTERPOLATION ON AX=Y IN A TRIDIAGONAL ALGEBRA ALG${\pounds}$

  • Kang, Joo-Ho
    • The Pure and Applied Mathematics
    • /
    • v.15 no.4
    • /
    • pp.401-406
    • /
    • 2008
  • Given operators X and Y acting on a separable complex Hilbert space H, an interpolating operator is a bounded operator A such that AX=Y. In this article, we investigate Hilbert-Schmidt interpolation problems for operators in a tridiagonal algebra and we get the following: Let ${\pounds}$ be a subspace lattice acting on a separable complex Hilbert space H and let X=$(x_{ij})$ and Y=$(y_{ij})$ be operators acting on H. Then the following are equivalent: (1) There exists a Hilbert-Schmidt operator $A=(a_{ij})$ in Alg${\pounds}$ such that AX=Y. (2) There is a bounded sequence $\{{\alpha}_n\}$ in $\mathbb{C}$ such that ${\sum}_{n=1}^{\infty}|{\alpha}_n|^2<{\infty}$ and $$y1_i={\alpha}_1x_{1i}+{\alpha}_2x_{2i}$$ $$y2k_i={\alpha}_{4k-1}x_2k_i$$ $$y{2k+1}_i={\alpha}_{4k}x_{2k}_i+{\alpha}_{4k+1}x_{2k+1}_i+{\alpha}_{4k+2}x_{2k+2}_i\;for\;all\;i,\;k\;\mathbb{N}$$.

  • PDF

MULTIPLIERS FOR OPERATOR-VALUED BESSEL SEQUENCES AND GENERALIZED HILBERT-SCHMIDT CLASSES

  • KRISHNA, K. MAHESH;JOHNSON, P. SAM;MOHAPATRA, R.N.
    • Journal of applied mathematics & informatics
    • /
    • v.40 no.1_2
    • /
    • pp.153-171
    • /
    • 2022
  • In 1960, Schatten studied operators of the form $\sum_{n=1}^{{\infty}}\;{\lambda}_n(x_n{\otimes}{\bar{y_n}})$, where {xn}n and {yn}n are orthonormal sequences in a Hilbert space, and {λn}n ∈ ℓ(ℕ). Balazs generalized some of the results of Schatten in 2007. In this paper, we further generalize results of Balazs by studying the operators of the form $\sum_{n=1}^{{\infty}}\;{\lambda}_n(A^*_nx_n{\otimes}{\bar{B^*_ny_n}})$, where {An}n and {Bn}n are operator-valued Bessel sequences, {xn}n and {yn}n are sequences in the Hilbert space such that {║xn║║yn║}n ∈ ℓ(ℕ). We also generalize the class of Hilbert-Schmidt operators studied by Balazs.

AN EXTENSION OF THE FUGLEDGE-PUTNAM THEOREM TO $\omega$-HYPONORMAL OPERATORS

  • Cha, Hyung Koo
    • The Pure and Applied Mathematics
    • /
    • v.10 no.4
    • /
    • pp.273-277
    • /
    • 2003
  • The Fuglede-Putnam Theorem is that if A and B are normal operators and X is an operator such that AX = XB, then $A^{\ast}= X. In this paper, we show that if A is $\omega$-hyponormal and $B^{\ast}$ is invertible $\omega$-hyponormal such that AX = XB for a Hilbert-Schmidt operator X, then $A^{\ast}X = XB^{\ast}$.

  • PDF

AN EXTENSION OF THE FUGLEDE-PUTNAM THEOREM TO k-QUASIHYPONORMAL OPERATORS

  • Shin, Kyo-Il;Cha, Hyung-Koo
    • East Asian mathematical journal
    • /
    • v.14 no.1
    • /
    • pp.21-26
    • /
    • 1998
  • The Fulgede-Putnam theorem asserts as if A and Bare normal operators and X is an operator such that AX=XB, then A*X=XB*. In this paper, we show that if A is k-quasihyponormal and B* is invertible k-quasihyponormal such that AX=XB for a Hilbert-Schmidt operator X, then A*X=XB*.

  • PDF

A BERBERIAN TYPE EXTENSION OF FUGLEDE-PUTNAM THEOREM FOR QUASI-CLASS A OPERATORS

  • Kim, In Hyoun;Jeon, In Ho
    • Korean Journal of Mathematics
    • /
    • v.16 no.4
    • /
    • pp.583-587
    • /
    • 2008
  • Let $\mathfrak{L(H)}$ denote the algebra of bounded linear operators on a separable infinite dimensional complex Hilbert space $\mathfrak{H}$. We say that $T{\in}\mathfrak{L(H)}$ is a quasi-class A operator if $$T^*{\mid}T^2{\mid}T{{\geq}}T^*{\mid}T{\mid}^2T$$. In this paper we prove that if A and B are quasi-class A operators, and $B^*$ is invertible, then for a Hilbert-Schmidt operator X $$AX=XB\;implies\;A^*X=XB^*$$.

  • PDF

SOME TRACE INEQUALITIES FOR CONVEX FUNCTIONS OF SELFADJOINT OPERATORS IN HILBERT SPACES

  • Dragomir, Silvestru Sever
    • Korean Journal of Mathematics
    • /
    • v.24 no.2
    • /
    • pp.273-296
    • /
    • 2016
  • Some new trace inequalities for convex functions of self-adjoint operators in Hilbert spaces are provided. The superadditivity and monotonicity of some associated functionals are investigated. Some trace inequalities for matrices are also derived. Examples for the operator power and logarithm are presented as well.