• Kyungpook Mathematical Journal
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• v.63 no.2
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• pp.235-250
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• 2023

Let 1 ≤ p, q < ∞ be fixed, and let R = [r_jk] be an infinite scalar matrix such that 1 ≤ r_jk < ∞ and sup_j,k r_jk < ∞. Let 𝓑(𝑙_p, 𝑙_q) be the set of all bounded linear operator from 𝑙_p into 𝑙_q. For a fixed Banach algebra 𝐁 with identity, we define a new vector space S^R_p,q(𝐁) of infinite matrices over 𝐁 and a paranorm G on S^R_p,q(𝐁) as follows: let $$S^R_{p,q}({\mathbf{B}})=\{A:A^{[R]}{\in}{\mathcal{B}}(l_p,l_q)\}$$ and $G(A)={\parallel}A^{[R]}{\parallel}^{\frac{1}{M}}_{p,q}$, where $A^{[R]}=[{\parallel}a_{jk}{\parallel}^{r_{jk}}]$ and M = max{1, sup_j,k r_jk}. The existance of S^R_p,q(𝐁) equipped with the paranorm G(·) including its completeness are studied. We also provide characterizations of β -dual of the paranormed space.

• The Pure and Applied Mathematics
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• v.7 no.2
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• pp.79-86
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• 2000

In this paper, we introduce the classes H(p,q,k),K(p;k) of operators determined by the Heinz-Kato-Furuta inequality and Holer-McCarthy inequality. We characterize relationship between p-quasihyponormal, $\kappa$-quasihyponormal and $\kappa$-p-quasihyponormal operators. And it is proved that every operator in K(p;1) for some $0 is paranormal.

• Communications of the Korean Mathematical Society
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• v.24 no.1
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• pp.57-66
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• 2009

In this paper, we characterize the boundedness and compactness of the extended $Ces{\grave{a}}ro$ operators from general function spaces F(p, q, s) to Bloch-type spaces ${\mathcal{B}}_{\mu}$ where $\mu$ is normal function on [0,1).

• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1255-1268
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• 2022

We consider the Schrödinger type operator 𝓛 = (-𝚫_ℍⁿ)² + V² on the Heisenberg group ℍⁿ, where 𝚫_ℍⁿ is the sub-Laplacian and the non-negative potential V belongs to the reverse Hölder class RH_s for s ≥ Q/2 and Q ≥ 6. We shall establish the (L^p, L^q) estimates for the Riesz transforms T_𝛼,𝛽,j = V^2𝛼𝛁^j_ℍⁿ𝓛^-𝛽, j = 0, 1, 2, 3, where 𝛁_ℍⁿ is the gradient operator on ℍⁿ, 0 < α ≤ 1-j/4, j/4 < 𝛽 ≤ 1, and 𝛽 - 𝛼 ≥ j/4.

• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1117-1127
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• 2019

Let ${\mathcal{L}}_2=(-{\Delta})^2+V^2$ be the $Schr{\ddot{o}}dinger$ type operator, where nonnegative potential V belongs to the reverse $H{\ddot{o}}lder$ class $RH_s$, s > n/2. In this paper, we consider the operator $T_{{\alpha},{\beta}}=V^{2{\alpha}}{\mathcal{L}}^{-{\beta}}_2$ and its conjugate $T^*_{{\alpha},{\beta}}$, where $0<{\alpha}{\leq}{\beta}{\leq}1$. We establish the $(L^p,\;L^q)$-boundedness of operator $T_{{\alpha},{\beta}}$ and $T^*_{{\alpha},{\beta}}$, respectively, we also show that $T_{{\alpha},{\beta}}$ is bounded from Hardy type space $H^1_{L_2}({\mathbb{R}}^n)$ into $L^{p_2}({\mathbb{R}}^n)$ and $T^*_{{\alpha},{\beta}}$ is bounded from $L^{p_1}({\mathbb{R}}^n)$ into BMO type space $BMO_{{\mathcal{L}}1}({\mathbb{R}}^n)$, where $p_1={\frac{n}{4({\beta}-{\alpha})}}$, $p_2={\frac{n}{n-4({\beta}-{\alpha})}}$.

• Communications of the Korean Mathematical Society
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• v.28 no.1
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• pp.1-9
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• 2013

We consider the sets of matrix ordered pairs which satisfy extremal properties with respect to Boolean rank inequalities of matrices over nonbinary Boolean algebra. We characterize linear operators that preserve these sets of matrix ordered pairs as the form of $T(X)=PXP^T$ with some permutation matrix P.

• The Pure and Applied Mathematics
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• v.14 no.2 s.36
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• pp.127-137
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• 2007

The classical Bohr's inequality states that $|z+w|^2{\leq}p|z|^2+q|w|^2$ for all $z,\;w{\in}\mathbb{C}$ and all p, q>1 with $\frac{1}{p}+\frac{1}{q}=1$. In this paper, Bohr's inequality is generalized to the setting of n-inner product spaces for all positive conjugate exponents $p,\;q{\in}\mathbb{R}$. In. In particular, the parallelogram law is recovered and an interesting operator inequality is obtained.

• Journal of the Korean Mathematical Society
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• v.40 no.6
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• pp.933-942
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• 2003

It is shown that if a paranormal contraction T has no nontrivial invariant subspace, then it is a proper contraction. Moreover, the nonnegative operator Q = T/sup 2*/T/sup 2/ - 2T/sup */T ＋ I also is a proper contraction. If a quasihyponormal contraction has no nontrivial invariant subspace then, in addition, its defect operator D is a proper contraction and its itself-commutator is a trace-class strict contraction. Furthermore, if one of Q or D is compact, then so is the other, and Q and D are strict ontraction.

• Journal of the Korean Mathematical Society
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• v.45 no.2
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• pp.301-312
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• 2008

The spanning column rank of an $m{\times}n$ matrix A over a semiring is the minimal number of columns that span all columns of A. We characterize linear operators that preserve the sets of matrix pairs which satisfy additive properties with respect to spanning column rank of matrices over semirings.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1071-1085
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• 2016

For $b{\in}L^1_{loc}({\mathbb{R}}^n)$, let ${\mathcal{I}}_{\alpha}$ be the bilinear fractional integral operator, and $[b,{\mathcal{I}}_{\alpha}]_i$ be the commutator of ${\mathcal{I}}_{\alpha}$ with pointwise multiplication b (i = 1, 2). This paper shows that if the commutator $[b,{\mathcal{I}}_{\alpha}]_i$ for i = 1 or 2 is bounded from the product Morrey spaces $L^{p_1,{\lambda}_1}({\mathbb{R}}^n){\times}L^{p_2,{\lambda}_2}({\mathbb{R}}^n)$ to the Morrey space $L^{q,{\lambda}}({\mathbb{R}}^n)$ for some suitable indexes ${\lambda}$, ${\lambda}_1$, ${\lambda}_2$ and $p_1$, $p_2$, q, then $b{\in}BMO({\mathbb{R}}^n)$, as well as that the compactness of $[b,{\mathcal{I}}_{\alpha}]_i$ for i = 1 or 2 from $L^{p_1,{\lambda}_1}({\mathbb{R}}^n){\times}L^{p_2,{\lambda}_2}({\mathbb{R}}^n)$ to $L^{q,{\lambda}}({\mathbb{R}}^n)$ implies that $b{\in}CMO({\mathbb{R}}^n)$ (the closure in $BMO({\mathbb{R}}^n)$of the space of $C^{\infty}({\mathbb{R}}^n)$ functions with compact support). These results together with some previous ones give a new characterization of $BMO({\mathbb{R}}^n)$ functions or $CMO({\mathbb{R}}^n)$ functions in essential ways.