• 한국수학교육학회지시리즈B:순수및응용수학
• /
• 제21권2호
• /
• pp.121-128
• /
• 2014

The Boolean rank of a nonzero m $m{\times}n$ Boolean matrix A is the least integer k such that there are an $m{\times}k$ Boolean matrix B and a $k{\times}n$ Boolean matrix C with A = BC. In 1984, Beasley and Pullman showed that a linear operator preserves the Boolean rank of any Boolean matrix if and only if it preserves Boolean ranks 1 and 2. In this paper, we extend this characterization of linear operators that preserve the Boolean ranks of Boolean matrices. We show that a linear operator preserves all Boolean ranks if and only if it preserves two consecutive Boolean ranks if and only if it strongly preserves a Boolean rank k with $1{\leq}k{\leq}min\{m,n\}$.

• Kyungpook Mathematical Journal
• /
• 제48권2호
• /
• pp.281-286
• /
• 2008

In this note we prove that if A and $B^*$ are subnormal operators and is a bounded linear operator such that AX - XB is a Hilbert-Schmidt operator, then f(A)X - Xf(B) is also a Hilbert-Schmidt operator and $${\parallel}f(A)X\;-\;Xf(B){\parallel}_2\;\leq\;L{\parallel}AX\;-\;XB{\parallel}_2$$, for f belonging to a certain class of functions. Furthermore, we investigate the similar problem in the case that S, T are hyponormal operators and $X\;{\in}\;\cal{L}(\cal{H})$ is such that SX - XT belongs to a norm ideal (J, ${\parallel}\;{\cdot}\;{\parallel}_J$) and prove that f(S)X - Xf(T) $\in$ J and ${\parallel}f(S)X\;-\;Xf(T){\parallel}_J\;\leq\;C{\parallel}SX\;-\;XT{\parallel}_J$, for f in a certain class of functions.

• Journal of applied mathematics & informatics
• /
• 제26권3_4호
• /
• pp.457-469
• /
• 2008

In this paper, we study the existence of positive solutions for the following nonlinear m-point boundary value problem with p-Laplacian: $\{{{{(\phi_p(u'))'\;+\;f(t,u(t))=0, \;0<t<1,} \atop u'(0)={\sum}{^{m-2}_{i=1}}\;a_iu'(\xi_i),} \atop u(1)={\sum}{^k_{i=1}}\;b_iu(\xi_i)\;-\;{\sum}{^s_{i=k+1}}\;b_iu(\xi_i)\;-\;{\sum}{^{m-2}_{i=s+1}}\;b_iu'(xi_i),}$ where ${\phi}_p(s)$ is p-Laplacian operator, i.e., ${\phi}_p(s)=\mid s\mid^{p-2}s$, p>1, ${\phi}_q\;=\;({\phi}_p)^{-1}$, $\frac{1}{p}+\frac{1}{q}=1$, $1\;{\leq}\;k\;{\leq}\;s\;{\leq}m\;-\;2$, $b_i\;{\in}\;(0,+{\infty})$ with $0\;<\;{\sum}{^k_{k=1}}\;b_i\;-\;{\sum}{^s_{i=k+1}}\;b_i\;<\;1$, $0\;<\;{\sum}{^{m-2}_{i=1}}\la_i\;<\;1$, $0\;<\;{\xi}_1\;<\;{\xi}_2\;<\;{\cdots}\;<\;{\xi}_{m-2}\;<\;1$, $f\;{\in}\;C([0,\;1]\;{\times}\;[0,\;+{\infty}),\;[0,\;+{\infty}))$. We show that there exists one or two positive solutions by using fixed-point theorem for operator on a cone. The conclusions in this paper essentially extend and improve the known results.

• 대한수학회논문집
• /
• 제33권3호
• /
• pp.889-899
• /
• 2018

In this paper, we present some upper bounds for unitarily invariant norms inequalities. Among other inequalities, we show some upper bounds for the Hilbert-Schmidt norm. In particular, we prove $${\parallel}f(A)Xg(B){\pm}g(B)Xf(A){\parallel}_2{\leq}{\Large{\parallel}}{\frac{(I+{\mid}A{\mid})X(I+{\mid}B{\mid})+(I+{\mid}B{\mid})X(I+{\mid}A{\mid})}{^dA^dB}}{\Large{\parallel}}_2$$, where A, B, $X{\in}{\mathbb{M}}_n$ such that A, B are Hermitian with ${\sigma}(A){\cup}{\sigma}(B){\subset}{\mathbb{D}}$ and f, g are analytic on the complex unit disk ${\mathbb{D}}$, g(0) = f(0) = 1, Re(f) > 0 and Re(g) > 0.

• 대한수학회지
• /
• 제51권2호
• /
• pp.345-362
• /
• 2014

We describe the Haagerup tensor product ${\ell}^{\infty}{\otimes}_h{\ell}^{\infty}$ and the extended Haagerup tensor product ${\ell}^{\infty}{\otimes}_{eh}{\ell}^{\infty}$ in terms of Schur product maps, and show that ${\ell}^{\infty}{\otimes}_h{\ell}^{\infty}{\cap}\mathbb{B}({\ell}^2)$(resp. ${\ell}^{\infty}{\otimes}_{eh}{\ell}^{\infty}{\cap}\mathbb{B}({\ell}^2)$) coincides with $c_0{\otimes}_hc_0{\cap}\mathbb{B}({\ell}^2)$(resp. $c_0{\otimes}_{eh}c_0{\cap}\mathbb{B}({\ell}^2)$). For $C^*2$-algebras A, B, it is shown that $A{\otimes}_hB=A{\otimes}_{eh}B$ if and only if A or B is finite-dimensional.

• 대한수학회보
• /
• 제61권4호
• /
• pp.949-957
• /
• 2024

Let Y_p be the multiplication operator M_p plus the Volterra operator V_p induced by p(z), where p is a polynomial. Under a mild condition, we prove that Y_p acting on the Dirichlet space 𝔇 is similar to multiplication operator M_p acting on a subspace S(𝔻) of 𝔇. Furthermore, it shows that T_zⁿ (n ≥ 2) has exactly 2ⁿ reducing subspaces on 𝔇.

• 대한수학회보
• /
• 제55권4호
• /
• pp.1209-1219
• /
• 2018

We provide a direct proof of the following theorem of Kalton, Hollenbeck, and Verbitsky [7]: let H be the Hilbert transform and let a, b be real constants. Then for 1 < p < ${\infty}$ the norm of the operator aI + bH from $L^p(\mathbb{R})$ to $L^p(\mathbb{R})$ is equal to $$\({\max_{x{\in}{\mathbb{R}}}}{\frac{{\mid}ax-b+(bx+a){\tan}{\frac{\pi}{2p}}{\mid}^p+{\mid}ax-b-(bx+a){\tan}{\frac{\pi}{2p}}{\mid}^p}{{\mid}x+{\tan}{\frac{\pi}{2p}}{\mid}^p+{\mid}x-{\tan}{\frac{\pi}{2p}}{\mid}^p}}\)^{\frac{1}{p}}$$. Our proof avoids passing through the analogous result for the conjugate function on the circle, as in [7], and is given directly on the line. We also provide new approximate extremals for aI + bH in the case p > 2.

• 대한수학회지
• /
• 제45권3호
• /
• pp.771-780
• /
• 2008

Let $M_C=\(\array{A&C\\0&B}\)$ be a $2{\times}2$ upper triangular operator matrix acting on the Hilbert space $H{\bigoplus}K\;and\;let\;{\sigma}_w(\cdot)$ denote the Weyl spectrum. We give the necessary and sufficient conditions for operators A and B which ${\sigma}_w\(\array{A&C\\0&B}\)={\sigma}_w\(\array{A&C\\0&B}\)\;or\;{\sigma}_w\(\array{A&C\\0&B}\)={\sigma}_w(A){\cup}{\sigma}_w(B)$ holds for every $C{\in}B(K,\;H)$. We also study the Weyl's theorem for operator matrices.

• East Asian mathematical journal
• /
• 제31권5호
• /
• pp.695-706
• /
• 2015

Let ${\mathcal{H}}$ be an infinite dimensional complex Hilbert space, and let $B({\mathcal{H}})$ be the algebra of all bounded linear operators on ${\mathcal{H}}$. Recall that an operator $T{\in}B({\mathcal{H})$ has property B(n) if ${\mid}T^n{\mid}{\geq}{\mid}T{\mid}^n$, $n{\geq}2$, which generalizes the class A-operator. We characterize the property B(n) of weighted shifts $S_{\lambda}$ over (${\eta},\;{\kappa}$)-type directed trees which appeared in the study of subnormality of weighted shifts over directed trees recently. In addition, we discuss the property B(n) of weighted shifts $S_{\lambda}$ over (2, 1)-type directed trees with nonzero weights are being distinct with respect to $n{\geq}2$. And we give some properties of weighted shifts $S_{\lambda}$ over (2, 1)-type directed trees with property B(2).

• Kyungpook Mathematical Journal
• /
• 제49권4호
• /
• pp.595-603
• /
• 2009

We show that for an algebraic Riccati equation $X^*B^{-1}X-T^*X-X^*T=C$, its solutions are given by X = W + BT for some solution W of $X^*B^{-1}X$ = $C+T^*BT$. To generalize this, we give an equivalent condition for $\(\array{B&W\\W*&A}\)\;{\geq}\;0$ for given positive operators B and A, by which it can be regarded as Riccati inequality $X^*B^{-1}X{\leq}A$. As an application, the harmonic mean B ! C is explicitly written even if B and C are noninvertible.