Every Invertible Hilbert Space Operator is a Product of Seven Positive Operators

“…We note that Wu's result was improved by N. C. Phillips-he shows that every unitary on an infinite-dimensional Hilbert space is a product of six positive operators [4]. However, our techniques do not appear to be sufficient to obtain this stronger result.…”

Section: Theorem (P Y Wu) Let U Be a Unitary Operator On An Infinimentioning

confidence: 83%

“…It therefore suffices to show that any such matrix in M 2 (C) is the product of two positive matrices. Clearly, such a matrix is similar to a diagonal positive matrix p, so it is of the form vpv −1 , for some invertible matrix v. 4 do not depend on n-they are just four positive factors of the matrix b that occurs in the proof of the preceding lemma). Now form operator direct sums P 1 , P 2 , P 3 , P 4 on H = (K ⊕K)⊕(K ⊕K)⊕ · · · in such a way that P j has constant diagonal entries p j .…”

mentioning

confidence: 99%

Products of positive operators

Murphy

1997

Proc. Amer. Math. Soc.

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Abstract. A new, very simple proof is given of a result of P. Y. Wu which asserts that every unitary operator on an infinite-dimensional Hilbert space is a product of positive operators.A number of mathematicians have considered the problem of writing an operator as a product of "nice" operators, such as positive, hermitian or normal operators. Our principal reference for this is a paper of P. Y. Wu [6], but see also [2] and [5]. This kind of question, and related questions, have also been considered in a C * -algebra context, see [3]. A core result of Wu's paper is his theorem that a unitary operator on an infinitedimensional Hilbert space is a product of (sixteen) positive operators. This is an unexpected result, given what occurs in finite dimensions. For, in the latter situation, if we make the usual identification of an operator with a finite square matrix, a theorem of C. S. Ballantine [1] asserts that a matrix is a product of positive matrices precisely when its determinant is non-negative.The aim of this paper is to present a new proof of Wu's unitary result that is simpler and more perspicuous. Our version is so simple that we do not even have to use Ballantine's theorem (Wu's proof of his unitary result does make use of this theorem). In fact, we derive a weak version of Ballantine's theorem from our methods that has the merit of also being very simple and perspicuous. Lemma. Let u be a unitary element of a unital C* -algebra A. Then the matrixProof. To see this, we need only show that the scalar matrixis the product of four positive scalar matrices. For, in this case, using the fact that the matrix c = 0 1 u 0

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Products of Positive Operators

Contino

Dritschel

Maestripieri

et al. 2021

Complex Anal. Oper. Theory

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On finite dimensional spaces, it is apparent that an operator is the product of two positive operators if and only if it is similar to a positive operator. Here, the class $${\mathcal {L}^{+\,2}}$$ L + 2 of bounded operators on separable infinite dimensional Hilbert spaces which can be written as the product of two bounded positive operators is studied. The structure is much richer, and connects (but is not equivalent to) quasi-similarity and quasi-affinity to a positive operator. The spectral properties of operators in $${\mathcal {L}^{+\,2}}$$ L + 2 are developed, and membership in $${\mathcal {L}^{+\,2}}$$ L + 2 among special classes, including algebraic and compact operators, is examined.

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Every Invertible Hilbert Space Operator is a Product of Seven Positive Operators

Cited by 6 publications

References 6 publications

The semigroup generated by a similarity orbit or a unitary orbit of an operator

The semigroup generated by a similarity orbit or a unitary orbit of an operator

Products of positive operators

Products of Positive Operators

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