Trace class perturbations of isometries and unitary dilations
Trace class perturbations of isometries and unitary dilations
复制标题
等距和酉膨胀的迹类扰动
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发表时间：
1974
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通讯作者：
R. Carey
中科院分区：
文献类型：
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作者：
R. Carey
设U和V是作用于希尔伯特空间的等距线。如果uv属于迹类，则证明了相应的最小酉扩张的绝对连续部分是酉等价的。介绍。对于X和Y复希尔伯特空间，设fB（X， Y）表示从X到Y的有界线性算子集合；符号T(X)将表示fB（X， X）中的跟踪类元素。如果A是一个(不一定是有界)selfad联合运营商X = fR X dEA和谱表示X C X,然后{tl (^) = IE (A) X | K是一个非负,完全为波莱尔集定义附加式度量A .如果这个测量是绝对连续对勒贝格测度,我们说X是绝对连续对A集合(一)交流的所有向量X的衡量πj)是绝对连续是一个封闭的子空间减少[4],以及A对这个空间的限制，aac被称为A的绝对连续部分，类似的定义也适用于酉算子a1。关于连续谱摄动的一个重要结果是Rosenblum[1]和Kato[2]给出的，后来由Kuroda[3]推广。这个定理断言，如果IA， B}是Hilbert空间X中的一对自伴随算子，使得（a z） 1 (B z) Ic ET(X)对于某些z， imz a 0，则Aac是酉等价于B的。通过进行Cayley变换，这可以用酉算子来表述，即，如果U和V是酉算子，其中U V C J(X)，则Uac和Vac是酉等价的。本文的目的是证明这一事实对等距线对的推广。确切地说，我们证明了以下定理：设U和V是X中具有最小酉扩张的等距线。AMS （MOS）学科分类（1970年）。初级47A55, 47A20。
Let U and V be isometries acting on a Hilbert space. If U V belongs to the trace class we show that the absolutely continuous parts of the corresponding minimal unitary dilations are unitarily equivalent. Introduction. With X and Y complex Hilbert spaces let fB(X, Y) denote the collection of bounded linear operators from X to Y; the symbol T(X) will represent the trace class elements in fB(X, X). If A is a (not necessarily bounded) selfad joint operator in X with the spectral representation A = fR X dEA and x C X, then {tl(^) =IE(A)x|K is a nonnegative, completely additive measure defined for Borel sets A. If this measure is absolutely continuous with respect to Lebesgue measure, we say that x is absolutely continuous with respect to A. The collection X(A)ac of all vectors x for which the measure pi j) is absolutely continuous is a closed subspace which reduces A [4], and the restriction of A to this space, A ac is called the absolutely continuous part of A. A similar definition applies to a unitary operator A. 1. An important result on the perturbation of continuous spectra was given by Rosenblum [1] and Kato [2] and later generalized by Kuroda [3]. This theorem asserts that if IA, B} is a pair of selfadjoint operators in a Hilbert space X such that (A z) 1 (B z) Ic ET(X) for some z with Im z A 0, then Aac is unitarily equivalent to B . By taking Cayley transforms this may be rephrased in terms of unitary operators, i.e., if U and V are unitary operators for which U V C J(X), then Uac and Vac are unitarily equivalent. The object of this note is to prove an extension of this fact for pairs of isometries. Precisely, we prove the following: Theorem. Let U and V be isometries in X with minimal unitary dilations Received by the editors December 19, 1972. AMS (MOS) subject classifications (1970). Primary 47A55, 47A20.