• Journal of the Korean Mathematical Society
• /
• v.32 no.4
• /
• pp.775-784
• /
• 1995

Let $H$ be a separable, infnite dimensional, complex Hilbert space and let $L(H)$ denote the algebra of all bounded linear operators on $H$.

• Journal of the Chungcheong Mathematical Society
• /
• v.5 no.1
• /
• pp.159-165
• /
• 1992

In 1986, R. Huff [3] showed that a Dunford integrable function is Pettis integrable if and only if T : $X^*{\rightarrow}L_1(\mu)$ is weakly compact operator and {$T(K(F,\varepsilon))|F{\subset}X$, F : finite and ${\varepsilon}$ > 0} = {0}. In this paper, we introduce the notion of Bourgain property of real valued functions formulated by J. Bourgain [2]. We show that the class of pettis integrable functions is linear space and if lis bounded function with Bourgain property, then T : $X^{**}{\rightarrow}L_1(\mu)$ by $T(x^{**})=x^{**}f$ is $weak^*$ - to - weak linear operator. Also, if operator T : $L_1(\mu){\rightarrow}X^*$ with Bourgain property, then we show that f is Pettis representable.

• Korean Journal of Mathematics
• /
• v.22 no.1
• /
• pp.85-89
• /
• 2014

A bounded linear Hilbert space operator T is said to be k-quasi-class A operator if it satisfy the operator inequality $T^{*k}{\mid}T^2{\mid}T^k{\geq}T^{*k}{\mid}T{\mid}^2T^k$ for a non-negative integer k. It is proved that if T is a k-quasi-class A contraction, then either T has a nontrivial invariant subspace or T is a proper contraction and the nonnegative operator $D=T^{*k}({\mid}T^2{\mid}-{\mid}T{\mid}^2)T^k$ is strongly stable.

• Communications of the Korean Mathematical Society
• /
• v.9 no.3
• /
• pp.617-627
• /
• 1994

Algebraic spectral subspaces and admissible operators were introduced by K. B. Laursen and M. M. Neumann in 1988 [L88], [N]. These concepts are useful in automatic continuity problems of intertwining linear operators on Banach spaces. In this paper we characterize the algebraic spectral subspaces of generalized scalar operators. From this characterization we show that generalized scalar operators are admissible. Also we show that doubly power bounded operators are generalized scalar. And using the spectral capacity we show that a generalized scalar operator is decomposable. Then we give an example of an operator which is not admissible but decomposable.

• Communications of the Korean Mathematical Society
• /
• v.9 no.2
• /
• pp.331-336
• /
• 1994

Let $P_{0}$ be a $\delta$-ring (a ring closed with respect to the forming of countable intersections) of subsets of a nonempty set $\Omega$. Let X and Y be Banach spaces and L(X, Y) the Banach space of all bounded linear operators from X to Y. A set function m : $P_{0}$ longrightarrow L(X, Y) is called an operator valued measure countably additive in the strong operator topology if for every x $\epsilon$ X the set function E longrightarrow m(E)x is a countably additive vector measure. From now on, m will denote an operator valued measure countably additive in the strong operator topology.(omitted)

• The Pure and Applied Mathematics
• /
• v.22 no.3
• /
• pp.275-283
• /
• 2015

Let T be a bounded linear operator on a complex Hilbert space H. For a positive integer k, an operator T is said to be a k-quasi-2-isometric operator if T^∗k(T^∗2T² − 2T^∗T + I)T^k = 0, which is a generalization of 2-isometric operator. In this paper, we consider basic structural properties of k-quasi-2-isometric operators. Moreover, we give some examples of k-quasi-2-isometric operators. Finally, we prove that generalized Weyl’s theorem holds for polynomially k-quasi-2-isometric operators.

• Bulletin of the Korean Mathematical Society
• /
• v.32 no.2
• /
• pp.205-209
• /
• 1995

Throughout this paper X is a Banach space and $\mu$ is the Lebesgue measure on [0, 1] and all operators are assumed to be bounded and linear. $L^1(\mu)$ is the Banach space of all (classes of) Lebesgue integrable functions on [0, 1] with its usual norm. Let $T : L^1(\mu) \to X$ be an operator.

• The Pure and Applied Mathematics
• /
• v.5 no.1
• /
• pp.23-32
• /
• 1998

In the present paper the author studies the decomposition property ($\delta$) of the bounded linear operators.

• Korean Journal of Mathematics
• /
• v.21 no.1
• /
• pp.75-80
• /
• 2013

Let T be a bounded linear operator on a complex Hilbert space $\mathfrak{H}$. An operator T is called class A operator if ${\mid}T^2{\mid}{\geq}{\mid}T{\mid}^2$ and is called class $A({\kappa})$ operator if $(T^*{\mid}T{\mid}^{2{\kappa}}T)^{\frac{1}{{\kappa}+1}}{\geq}{\mid}T{\mid}^2$ for a positive number ${\kappa}$. In this paper, we show that ${\sigma}$ is continuous when restricted to the set of class $A({\kappa})$ operators.

• Journal of the Korean Mathematical Society
• /
• v.47 no.2
• /
• pp.351-361
• /
• 2010

For a bounded linear operator T on a separable complex infinite dimensional Hilbert space $\mathcal{H}$, we say that T is a quasi-class (A, k) operator if $T^{*k}|T^2|T^k\;{\geq}\;T^{*k}|T|^2T^k$. In this paper we prove that if T is a quasi-class (A, k) operator and f is an analytic function on an open neighborhood of the spectrum of T, then f(T) satisfies Weyl's theorem. Also, we consider the tensor product for quasi-class (A, k) operators.