• Communications of the Korean Mathematical Society
• /
• v.33 no.3
• /
• pp.787-798
• /
• 2018

For $1{\leq}p{\leq}{\infty}$, let $H^p$ be the usual Hardy space on the unit circle. When ${\alpha}$ and ${\beta}$ are bounded functions, a singular integral operator $S_{{\alpha},{\beta}}$ is defined as the following: $S_{{\alpha},{\beta}}(f+{\bar{g}})={\alpha}f+{\beta}{\bar{g}}(f{\in}H^p,\;g{\in}zH^p)$. When p = 2, we study the hyponormality of $S_{{\alpha},{\beta}}$ when ${\alpha}$ and ${\beta}$ are some special functions.

• Communications of the Korean Mathematical Society
• /
• v.21 no.3
• /
• pp.451-464
• /
• 2006

In this paper, a sharp inequality for some multilineal singular integral operators are obtained. As the applications, we get the weighted $L^p(p>1)$ norm inequalities and L log L type estimate for the multilinear operators.

• Journal of the Chungcheong Mathematical Society
• /
• v.23 no.2
• /
• pp.305-313
• /
• 2010

In this paper, we prove that if $T{\in}L$(X) is a generalized scalar operator then Ker $T^p$ is the quasi-nilpotent part of T for some positive integer $p{\in}{\mathbb{N}}$. Moreover, we prove that a generalized scalar operator with finite spectrum is algebraic. In particular, a quasi-nilpotent generalized scalar operator is nilpotent.

• Journal of the Korean Mathematical Society
• /
• v.53 no.1
• /
• pp.147-159
• /
• 2016

In this paper, subordination results of analytic function $f{\in}{\mathcal{A}}_p$ involving linear operator ${\mathcal{K}}^{{\delta},{\lambda}}_{c,p}$ are obtained. By applying the differential subordination method, results are derived under some sufficient subordination conditions. On using some hypergeometric identities, corollaries of the main results are derived. Furthermore, convolution preserving properties for a class of multivalent analytic function associated with the operator ${\mathcal{K}}^{{\delta},{\lambda}}_{c,p}$ are investigated.

• Bulletin of the Korean Mathematical Society
• /
• v.34 no.2
• /
• pp.317-334
• /
• 1997

The existence of an operator-valued function space integral as an operator on $L_p(R) (1 \leq p \leq 2)$ was established for certain functionals which involved the Labesgue measure [1,2,6,7]. Johnson and Lapidus showed the existence of the integral as an operator on $L_2(R)$ for certain functionals which involved any Borel measures [5]. J. S. Chang and Johnson proved the existence of the integral as an operator from L_1(R)$ to $C_0(R)$ for certain functionals involving some Borel measures [3]. K. S. Chang and K. S. Ryu showed the existence of the integral as an operator from $L_p(R) to L_p'(R)$ for certain functionals involving some Borel measures [4].

• Journal of applied mathematics & informatics
• /
• v.28 no.3_4
• /
• pp.1003-1008
• /
• 2010

In allelic model $X\;=\;(x_1,\;x_2,\;\cdots,\;x_d)$, $$M_f(t)\;=\;f(p(t))\;-\;\int_0^t\;Lf(p(t))ds$$ is a P-martingale for diffusion operator L under the certain conditions. In this note, we define $T_tf\;=\;E_{p_0}^{p^*}\;[f((P(t))]$ for $t\;{\geq}\;0$ for using a new diffusion operator $L^*$ and we show the diffusion relations between $T_t$ and diffusion operator $L^*$.

• Communications of the Korean Mathematical Society
• /
• v.18 no.4
• /
• pp.653-659
• /
• 2003

Let $1{\;}\leq{\;}p{\;}\infty{\;}and{\;}{\alpha}{\;}>{\;}-1$. If f is a holomorphic self-map of the open unit disc U of C with f(0) = 0, then the quantity $\int_U\;\{\frac{$\mid$f'(z)$\mid$}{1\;-\;$\mid$f(z)$\mid$^2}\}^p\;(1\;-\;$\mid$z$\mid$)^{\alpha+p}dxdy$ is equivalent to the operator norm of the composition operator $C_f{\;}:{\;}B{\;}\rightarrow{\;}A^{p,{\alpha}$ defined by $C_fh{\;}={\;}h{\;}\circ{\;}f{\;}-{\;}h(0)$, where B and $A^{p,{\alpha}$ are the Bloch space and the weighted Bergman space on U respectively.

• Bulletin of the Korean Mathematical Society
• /
• v.43 no.4
• /
• pp.747-753
• /
• 2006

We say operators A, B on Hilbert space satisfy Fuglede-Putnam theorem if AX = X B for some X implies $A^*X=XB^*$. We show that if either (1) A is p-hyponormal and $B^*$ is a class y operator or (2) A is a class y operator and $B^*$ is p-hyponormal, then A, B satisfy Fuglede-Putnam theorem.

• Kyungpook Mathematical Journal
• /
• v.46 no.4
• /
• pp.553-563
• /
• 2006

Let A be a bounded linear operator acting on a Hilbert space H. The B-Weyl spectrum of A is the set ${\sigma}_{B{\omega}}(A)$ of all ${\lambda}{\in}\mathbb{C}$ such that $A-{\lambda}I$ is not a B-Fredholm operator of index 0. Let E(A) be the set of all isolated eigenvalues of A. Recently in [6] Berkani showed that if A is a hyponormal operator, then A satisfies generalized Weyl's theorem ${\sigma}_{B{\omega}}(A)={\sigma}(A)$\E(A), and the B-Weyl spectrum ${\sigma}_{B{\omega}}(A)$ of A satisfies the spectral mapping theorem. In [51], H. Weyl proved that weyl's theorem holds for hermitian operators. Weyl's theorem has been extended from hermitian operators to hyponormal and Toeplitz operators [12], and to several classes of operators including semi-normal operators ([9], [10]). Recently W. Y. Lee [35] showed that Weyl's theorem holds for algebraically hyponormal operators. R. Curto and Y. M. Han [14] have extended Lee's results to algebraically paranormal operators. In [19] the authors showed that Weyl's theorem holds for algebraically p-hyponormal operators. As Berkani has shown in [5], if the generalized Weyl's theorem holds for A, then so does Weyl's theorem. In this paper all the above results are generalized by proving that generalizedWeyl's theorem holds for the case where A is an algebraically ($p,\;k$)-quasihyponormal or an algebarically paranormal operator which includes all the above mentioned operators.

• Journal of the Korean Mathematical Society
• /
• v.45 no.4
• /
• pp.977-991
• /
• 2008

We characterize the boundedness and compactness of the weighted composition operator $uC_{\psi}$ from the general function space F(p, q, s) into the logarithmic Bloch space ${\beta}_L$ on the unit disk. Some necessary and sufficient conditions are given for which $uC_{\psi}$ is a bounded or a compact operator from F(p,q,s), $F_0$(p,q,s) into ${\beta}_L$, ${\beta}_L^0$ respectively.