• 대한수학회보
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• 제58권3호
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• pp.637-657
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• 2021

In this paper, we decompose (under unitary similarity) the Kronecker sum A ⊞ A (= A ⊗ I + I ⊗ A) into a direct sum of irreducible matrices, when A is a 3×3 matrix. As a consequence we identify 𝒦(A⊞A) as the direct sum of several full matrix algebras as predicted by Artin-Wedderburn theorem, where 𝒦(T) is the unital algebra generated by Tand T^*.

• 대한수학회보
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• 제54권1호
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• pp.215-223
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• 2017

In this paper, we establish results about operators similar to their adjoints. This is carried out in the setting of bounded and also unbounded operators on a Hilbert space. Among the results, we prove that an unbounded closed operator similar to its adjoint, via a cramped unitary operator, is self-adjoint. The proof of this result works also as a new proof of the celebrated result by Berberian on the same problem in the bounded case. Other results on similarity of hyponormal unbounded operators and their self-adjointness are also given, generalizing well known results by Sheth and Williams.

• 대한수학회보
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• 제35권1호
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• pp.91-97
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• 1998

In this paper we study the Weyl spectrum of weight $\alpha, \omega_\alpha(T)$, of an operator T acting on an infinite dimensional Hilbert space. Main results are as follows. Firstly, we show that the Weyll spectrum of weight $\alpha$ of a polynomially $\alpha$-compact operator is finite, and that similarity preserves polynomial $\alpha$-compactness and the $\alpha$-Weyl's theorem both. Secondly, we give a sufficient condition for an operator to be the sum of an unitary and a $\alpha$-compact operators.

• 호남수학학술지
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• 제31권3호
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• pp.421-428
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• 2009

Given operators X and Y acting on a Hilbert space $\mathcal{H}$, an interpolating operator is a bounded operator A such that AX = Y. In this paper, we showed the following : Let $\mathcal{L}$ be a subspace lattice acting on a Hilbert space $\mathcal{H}$ and let $X_i$ and $Y_i$ be operators in B($\mathcal{H}$) for i = 1, 2, ${\cdots}$. Let $P_i$ be the projection onto $\overline{rangeX_i}$ for all i = 1, 2, ${\cdots}$. If $P_kE$ = $EP_k$ for some k in $\mathbb{N}$ and all E in $\mathcal{L}$, then the following are equivalent: (1) $sup\;\{{\frac{{\parallel}E^{\perp}({\sum}^n_{i=1}Y_if_i){\parallel}}{{\parallel}E^{\perp}({\sum}^n_{i=1}Y_if_i){\parallel}}:f{\in}H,n{\in}{\mathbb{N}},E{\in}\mathcal{L}}\}$ < ${\infty}$ range $\overline{rangeY_k}\;=\;\overline{rangeX_k}\;=\;\mathcal{H}$, and < $X_kf,\;X_kg$ >=< $Y_kf,\;Y_kg$ > for some k in $\mathbb{N}$ and for all f and g in $\mathcal{H}$. (2) There exists an operator A in Alg$\mathcal{L}$ such that $AX_i$ = $Y_i$ for i = 1, 2, ${\cdots}$ and AA$^*$ = I = A$^*$A.

• 대한수학회지
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• 제38권2호
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• pp.337-348
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• 2001

The unitary Lie-Trotter-Kato product formula gives in a simplest way a meaning to the Feynman path integral for the Schroding-er equation. In this note we want to survey some of recent results on the norm convergence of the selfadjoint Lie-Trotter Kato product formula for the Schrodinger operator -1/2Δ + V(x) and for the sum of two selfadjoint operators A and B. As one of the applications, a remark is mentioned about an approximation therewith to the fundamental solution for the imaginary-time Schrodinger equation.

• 한국전자통신학회논문지
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• 제11권2호
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• pp.165-170
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• 2016

본 논문은 $NCV-{\mid}v_1$ > 라이브러리의 새로운 쌍대 구조의 특성과 그 응용에 대한 연구이다. 상태벡터가 고유벡터인 임의 n개 qudit 상의 모든 유니터리 연산은 1개와 2개의 $NCV-{\mid}v_1$ > 라이브러리 들의 합성으로 표현할 수 있다. 본 연구는 입력 상태벡터가 PP로 제한적인 Barenco의 n비트 U(2n) 연산자를 전역 극성의 입력 상태벡터로 확장 실현하는 것이다. 이와 같이 보강된 확장 실현은 유니터리 연산자의 제어게이트 합성 시에 $NCV-{\mid}v_1$ > 과 이의 쌍대인 $NCV^{\dag}-{\mid}v_1$ > 라이브러리 모두를 사용할 경우에 이들이 대칭적 쌍대 성질을 갖고 있어 모든 극성의 상태벡터 입력에 대해 AND 지배적 종속 합성이 가능하기 때문이다.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제16권3호
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• pp.277-281
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• 2009

A Hilbert space operator T is a 2-isometry if $T^{{\ast}2}T^2\;-\;2T^{\ast}T+I$ = O. We shall study some properties of 2-isometries, in particular spectra of a non-unitary 2-isometry and give an example. Also we prove with alternate argument that the Weyl's theorem holds for 2-isometries.

• Bulletin of the Korean Chemical Society
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• 제14권1호
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• pp.21-29
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• 1993

For the study of relaxation processes in complex spin system, a general master equation, which can be used to simulate a vast range of pulse experiments, has been formulated using the Liouville representation of quantum mechanics. The state of a nonequilibrium spin system in magnetic field is described by a density vector in Liouville space and the time evolution of the system is followed by the application of a linear master operator to the density vector in this Liouville space. In this master equation the nuclear spin relaxation due to intramolecular dipolar interaction or randomly fluctuating field interaction is explicitly implemented as a relaxation supermatrix for a strong coupled two-spin (1/2) system. The whole dynamic information inherent in the spin system is thus contained in the density vector and the master operator. The radiofrequency pulses are applied in the same space by corresponding unitary rotational supertransformations of the density vector. If the resulting FID is analytically Fourier transformed, it is possible to represent the final nonstationary spectrum using a frequency dependent spectral vector and intensity determining shape vector. The overall algorithm including relaxation interactions is then translated into an ANSIFORTRAN computer program, which can simulate a variety of two dimensional spectra. Furthermore a new strategy is tested by simulation of multiple quantum signals to differentiate the two relaxation interaction types.

• 호남수학학술지
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• 제31권4호
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• pp.479-487
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• 2009

Let $S(z_1,z_2),\;S_1(z_1,z_2)$ and $S_2(z_1,z_2)$ be power series with operator coefficients such that $S_(z_1,\;z_2)=S_1(z_1,z_2)S_2(z_1,z_2)$. Assume that the multiplications by $S_1(z_1,z_2)$ and $S_2(z_1,z_2)$ are contractive transformations in H($\mathbb{D}^2,\;\mathcal{C}$). Then the factorizations of spaces $\mathcal{D}(\mathbb{D},\;\tilde{S})$ and $\mathcal{D}(\mathbb{D}^2,\mathcal{S})$ are well-behaved.

• 한국전자통신학회논문지
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• 제12권6호
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• pp.1027-1034
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• 2017

본 논문은 양자게이트의 모든 제어 동작점에서 양자들의 확률진폭, 확률, 평균 기댓값 및 정상상태 단위행렬의 행렬요소 등을 수학적 투사로 측정할 수 있는 새로운 함수 임베딩 방법을 제안하였다. 본 논문의 함수 임베딩 방법은 디랙 기호와 크로네커델타 기호를 사용해 각 제어 동작점에 대한 확률진폭의 직교 정규화조건을 2진 스칼라 연산자에 임베딩 한 것이다. 이와 같은 함수 임베딩 방법은 양자게이트 함수를 단일양자들의 텐서 곱으로 표현하는 유니터리 변환에서 유니터리 게이트의 산술 멱함수 제어에 매우 효과적 수단임을 밝혔다. Ternary 2-qutrit cNOT 게이트에 본 논문이 제안한 함수 임베딩 방법을 적용했을 때의 진화연산과 투사측정 결과를 제시하고, 기존의 방법들과 비교 검토하였다.