• 호남수학학술지
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• 제33권1호
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• pp.19-25
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• 2011

Let G be the general linear group GL(n,$\mathbb{C}$), $W_0$ the Weyl group of G and W the extended a neWeyl group of G. Then it is well-known that W is a union of the double cosets $W_{0x}W_0$ as x moves over the set of dominant weights of W. It is also known that each double coset $W_{0x}W_0$ contains a unique element $m_x$ of the shortest length. These shortest length elements belong to what are called the canonical left cells. However, it is still an open problem to find the canonical left cell containing a given $m_x$. One of the mai purposes of this paper is to introduce a new approach to attack this question. In particular, we will present a conjecture which explicitly describes the canonical left cells containing an element $m_x$. We will show that our conjecture is true for some specific types of $m_x$.

• 대한수학회지
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• 제40권1호
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• pp.129-167
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• 2003

In this paper we introduce new notions of Ricci-like tensor and many kind of curvature-like tensors such that concircular, projective, or conformal curvature-like tensors defined on semi-Riemannian manifolds. Moreover, we give some geometric conditions which are equivalent to the Codazzi tensor, the Weyl tensor, or the second Bianchi identity concerned with such kind of curvature-like tensors respectively and also give a generalization of Weyl's Theorem given in [18] and [19].

• 대한수학회지
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• 제57권4호
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• pp.893-913
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• 2020

Let 𝓢 be the collection of the operator matrices $\(\array{A&C\\Z&B}\)$ where the range of C is closed. In this paper, we study the properties of operator matrices in the class 𝓢. We first explore various local spectral relations, that is, the property (β), decomposable, and the property (C) between the operator matrices in the class 𝓢 and their component operators. Moreover, we investigate Weyl and Browder type spectra of operator matrices in the class 𝓢, and as some applications, we provide the conditions for such operator matrices to satisfy a-Weyl's theorem and a-Browder's theorem, respectively.

• 대한수학회논문집
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• 제27권2호
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• pp.327-339
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• 2012

In this paper we study $h$-projectively semisymmetric, ${\phi}$-pro-jectively semisymmetric, $h$-Weyl semisymmetric and ${\phi}$-Weyl semisym- metric non-Sasakian ($k$, ${\mu}$)-contact metric manifolds. In all the cases the manifold becomes an ${\eta}$-Einstein manifold. As a consequence of these results we obtain that if a 3-dimensional non-Sasakian ($k$, ${\mu}$)-contact metric manifold satisfies such curvature conditions, then the manifold reduces to an N($k$)-contact metric manifold.

• 대한수학회보
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• 제38권1호
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• pp.163-174
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• 2001

In this paper we introduce a notion of the $\chi-CLASS$ operators, which is a class including hyponormal operators and consider their spectral properties related to Weyl spectra.

• 대한수학회보
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• 제54권5호
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• pp.1669-1675
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• 2017

The group of automorphisms of the first Weyl algebra acts on commuting ordinary differential operators with polynomial coefficient. In this paper we prove that for fixed generic spectral curve of genus two the set of orbits is infinite.

• 대한수학회지
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• 제56권6호
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• pp.1489-1502
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• 2019

This paper describes when a partial isometry satisfies several weak normal properties. Topics treated include quasi-normality, subnormality, hyponormality, p-hyponormality (p > 0), w-hyponormality, paranormality, normaloidity, spectraloidity, the von Neumann property and Weyl's theorem.

• 대한수학회지
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• 제54권1호
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• pp.281-302
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• 2017

For a Banach space operator $A{\in}B(\mathcal{X})$, let ${\sigma}(A)$, ${\sigma}_a(A)$, ${\sigma}_w(A)$ and ${\sigma}_{aw}(A)$ denote, respectively, its spectrum, approximate point spectrum, Weyl spectrum and approximate Weyl spectrum. The operator A is polaroid (resp., left polaroid), if the points $iso{\sigma}(A)$ (resp., $iso{\sigma}_a(A)$) are poles (resp., left poles) of the resolvent of A. Perturbation by compact operators preserves neither SVEP, the single-valued extension property, nor the polaroid or left polaroid properties. Given an $A{\in}B(\mathcal{X})$, we prove that a sufficient condition for: (i) A+K to have SVEP on the complement of ${\sigma}_w(A)$ (resp., ${\sigma}_{aw}(A)$) for every compact operator $K{\in}B(\mathcal{X})$ is that ${\sigma}_w(A)$ (resp., ${\sigma}_{aw}(A)$) has no holes; (ii) A + K to be polaroid (resp., left polaroid) for every compact operator $K{\in}B(\mathcal{X})$ is that iso${\sigma}_w(A)$ = ∅ (resp., $iso{\sigma}_{aw}(A)$ = ∅). It is seen that these conditions are also necessary in the case in which the Banach space $\mathcal{X}$ is a Hilbert space.

• 대한수학회지
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• 제51권5호
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• pp.1089-1104
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• 2014

An operator $T{\in}L(H)$, is said to belong to k-quasi class $A_n^*$ operator if $$T^{*k}({\mid}T^{n+1}{\mid}^{\frac{2}{n+1}}-{\mid}T^*{\mid}^2)T^k{\geq}O$$ for some positive integer n and some positive integer k. First, we will see some properties of this class of operators and prove Weyl's theorem for algebraically k-quasi class $A_n^*$. Second, we consider the tensor product for k-quasi class $A_n^*$, giving a necessary and sufficient condition for $T{\otimes}S$ to be a k-quasi class $A_n^*$, when T and S are both non-zero operators. Then, the existence of a nontrivial hyperinvariant subspace of k-quasi class $A_n^*$ operator will be shown, and it will also be shown that if X is a Hilbert-Schmidt operator, A and $(B^*)^{-1}$ are k-quasi class $A_n^*$ operators such that AX = XB, then $A^*X=XB^*$. Finally, we will prove the spectrum continuity of this class of operators.

• 대한수학회보
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• 제57권1호
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• pp.207-218
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• 2020

Some modular representations of reflection groups related to Weyl groups are considered. The rational cohomology of the classifying space of a compact connected Lie group G with a maximal torus T is expressed as the ring of invariants, H*(BG; ℚ) ≅ H*(BT; ℚ)^W(G), which is a polynomial ring. If such Lie groups are locally isomorphic, the rational representations of their Weyl groups are equivalent. However, the integral representations need not be equivalent. Under the mod p reductions, we consider the structure of the rings, particularly for the Weyl group of symplectic groups Sp(n) and for the alternating groups A_n as the subgroup of W(SU(n)). We will ask if such rings of invariants are polynomial rings, and if each of them can be realized as the mod p cohomology of a space. For n = 3, 4, the rings under a conjugate of W(Sp(n)) are shown to be polynomial, and for n = 6, 8, they are non-polynomial. The structures of H*(BT^n-1; 𝔽_p)^A_n will be also discussed for n = 3, 4.