Semi-regular biorthogonal pairs and generalized Riesz bases
Semi-regular biorthogonal pairs and generalized Riesz bases
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半正则双正交对和广义 Riesz 基
DOI：
10.1063/1.4966927
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发表时间：
2016
影响因子：
1.3
通讯作者：
H. Inoue
中科院分区：
文献类型：
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作者：
H. Inoue
本文介绍了半正则双正交对的一般理论、广义Riesz基及其物理应用。这里我们处理希尔伯特空间H中的双正交序列{ϕn}和{ψn}，域D (φ) = {x∈H；∑k = 0∞(x |ϕk) 2 <∞}和D(ψ)= {x∈H;∑k = 0∞(x | ψ k) 2 <∞}和线性跨度dφ≡跨度{ϕn}和Dψ≡跨度{ψn}。如果双正交对（{ϕn}, {ψn}）在H中都是密集的，则称其为正则对；如果Dϕ和D（φ）或Dψ和D（ψ）在H中都是密集的，则称其为半正则对。在之前的一篇论文中[H。井上J.数学。我们已经证明，如果（{ϕn}, {ψn}）是正则双正交对，那么{ϕn}和{ψn}都是在Inoue和Takakura的工作中定义的广义Riesz基[J]。数学。物理学报，57,083505 (2016)[j]。在这里，我们将通过使用算符tφ，e, Te，ϕ， Tψ，e和Te，如果这对只是半正则的，则同样的结果成立，其中ψ由H中的标准正交基e和双正交对（{ϕn}, {ψn}）定义。F……
In this paper we introduce general theories of semi-regular biorthogonal pairs, generalized Riesz bases and its physical applications. Here we deal with biorthogonal sequences {ϕn} and {ψn} in a Hilbert space H , with domains D ( ϕ ) = { x ∈ H ; ∑ k = 0 ∞ ( x | ϕ k ) 2 < ∞ } and D ( ψ ) = { x ∈ H ; ∑ k = 0 ∞ ( x | ψ k ) 2 < ∞ } and linear spans Dϕ ≡ Span{ϕn} and Dψ ≡ Span{ψn}. A biorthogonal pair ({ϕn}, {ψn}) is called regular if both Dϕ and Dψ are dense in H , and it is called semi-regular if either Dϕ and D(ϕ) or Dψ and D(ψ) are dense in H . In a previous paper [H. Inoue, J. Math. Phys. 57, 083511 (2016)], we have shown that if ({ϕn}, {ψn}) is a regular biorthogonal pair then both {ϕn} and {ψn} are generalized Riesz bases defined in the work of Inoue and Takakura [J. Math. Phys. 57, 083505 (2016)]. Here we shall show that the same result holds true if the pair is only semi-regular by using operators Tϕ,e, Te,ϕ, Tψ,e, and Te,ψ defined by an orthonormal basis e in H and a biorthogonal pair ({ϕn}, {ψn}). F...