• Bulletin of the Korean Mathematical Society
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• v.46 no.3
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• pp.447-461
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• 2009

In this paper we give a non-existence theorem for Hopf real hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$ satisfying the condition that the structure Jacobi operator $R_{\xi}$ commutes with the 3-structure tensors ${\phi}_i$, i = 1, 2, 3.

• Nonlinear Functional Analysis and Applications
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• v.26 no.1
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• pp.71-81
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• 2021

The main object of this present paper is to study some majorization problems for certain classes of analytic functions defined by means of q-calculus operator associated with exponential function.

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1061-1067
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• 2013

In this paper, we study surfaces of revolution without parabolic points in 3-Euclidean space $\mathbb{R}^3$, satisfying the condition ${\Delta}^{II}G=f(G+C)$, where ${\Delta}^{II}$ is the Laplace operator with respect to the second fundamental form, $f$ is a smooth function on the surface and C is a constant vector. Our main results state that surfaces of revolution without parabolic points in $\mathbb{R}^3$ which satisfy the condition ${\Delta}^{II}G=fG$, coincide with surfaces of revolution with non-zero constant Gaussian curvature.

• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.237-252
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• 2014

In this paper, we give a necessary and sufficient condition for the hyponormality of the block Toeplitz operators $T_{\Phi}$, where ${\Phi}$ = $F+G^*$, F(z), G(z) are some matrix valued polynomials on the vector valued Bergman space $L^2_a(\mathbb{D},\mathbb{C}^n)$. We also show some necessary conditions for the hyponormality of $T_{F+G^*}$ with $F+G^*{\in}h^{\infty}{\otimes}M_{n{\times}n}$ on $L^2_a(\mathbb{D},\mathbb{C}^n)$.

• Proceedings of the Korean Society of Computational and Applied Mathematics Conference
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• 2003.09a
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• pp.4.1-4
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• 2003

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 AXi= Yi, for i = 1,2,...,n, In this article, we showed the following : Let H be a Hilbert space and let L be a subspace lattice on H. Let X and Y be operators acting on H. Assume that rangeX is dense in H. Then the following statements are equivalent : (1) There exists an operator A in AlgL such that AX = Y, A$\^$*/=A and every E in L reduces A. (2) sup｛(equation omitted) : n $\in$ N f$\sub$I/ $\in$ H and E$\sub$I/ $\in$ L｝<$\infty$ and = for all E in L and all f, g in H.

• Communications of the Korean Mathematical Society
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• v.37 no.1
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• pp.229-257
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• 2022

Let M be a semi-invariant submanifold of codimension 3 with almost contact metric structure (𝜙, 𝜉, 𝜂, g) in a complex space form M_n+1(c), c ≠ 0. We denote by A and R_𝜉 the shape operator in the direction of distinguished normal vector field and the structure Jacobi operator with respect to the structure vector 𝜉, respectively. Suppose that the third fundamental form t satisfies dt(X, Y) = 2𝜃g(𝜙X, Y) for a scalar 𝜃(< 2c) and any vector fields X and Y on M. In this paper, we prove that if it satisfies R_𝜉A = AR_𝜉 and at the same time ∇_𝜉R_𝜉 = 0 on M, then M is a Hopf hypersurface of type (A) provided that the scalar curvature s of M holds s - 2(n - 1)c ≤ 0.

• Bulletin of the Korean Mathematical Society
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• v.26 no.1
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• pp.11-17
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• 1989

In [1], Johnson and Lapidus introduced a family { $A_{t}$ :t>0} of Banach algebras of functionals on Wiener space and showed that for every F in $A_{t}$ , the analytic operator-valued function space integral $K_{\lambda}$$^{t}$ (F) exists for all nonzero complex numbers .lambda. with nonnegative real part. In [2,3] Johnson and Lapidus introduced a noncommtative multiplication having the property that if F.mem. $A_{t}$ $_{1}$ and G.mem. $A_{t}$ $_{2}$ then $F^{*}$G.mem. A$t_{1}$+$_{t}$ $_{2}$ and (Fig.) Note that for F, G in $A_{t}$ , $F^{*}$G is not in $A_{t}$ but rather is in $A_{2t}$ and so the multiplication * is not internal to the Banach algebra $A_{t}$ . In this paper we introduce an internal noncommutative multiplication on $A_{t}$ having the property that for F, G in $A_{t}$ , F G is in $A_{t}$ and (Fig.) for all nonzero .lambda. with nonnegative real part. Thus is an auxiliary binary operator on $A_{t}$ .TEX> .

• Bulletin of the Korean Mathematical Society
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• v.34 no.4
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• pp.627-636
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• 1997

Let $H(U)$ be the space of all analytic functions in the unit disk $U$ and let $K \subset H(U)$. For the operator $A_{\beta,\gamma} : K \longrightarrow H(U)$ defined by $$ A_{\beta,\gamma}(f)(z) = [\frac{z^\gamma}{\beta + \gamma} \int_{0}^{z} f^\beta (t)t^{\gamma-1} dt]^{1/\beta} $$ and $\beta,\gamma \in C$, we determined conditions on g(z), $\beta and \gamma$ such that $$ z[\frac{z}{f(z)]^\beta \prec z[\frac{z}{g(z)]^\beta implies z[\frac{z}{A_{\beta,\gamma}(f)(z)]^\beta \prec z[\frac{z}{A_{\beta,\gamma}(g)(z)]^\beta $$ and we presented some particular cases of our main result.

• Korean Journal of Mathematics
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• v.10 no.1
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• pp.11-17
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• 2002

In this paper we give several necessary and sufficient conditions for an operator on the Hilbert space to obey Browder's theorem. And it is shown that if S has totally finite ascent and $T{\prec}S$ then $f(T)$ obeys Browder's theorem for every $f{\in}H({\sigma}(T))$, where $H({\sigma}(T))$ denotes the set of all analytic functions on an open neighborhood of ${\sigma}(T)$.

• Kyungpook Mathematical Journal
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• v.61 no.1
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• pp.181-190
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• 2021

Let M be a real hypersurface with almost contact metric structure (ϕ, ��, η, g) in a nonflat complex space form Mn(c). We denote S and R�� by the Ricci tensor of M and by the structure Jacobi operator with respect to the vector field �� respectively. In this paper, we prove that M is a Hopf hypersurface of type A in Mn(c) if it satisfies R��ϕ = ϕR�� and at the same time R��(Sϕ - ϕS) = 0.