On the Spectrum of Unbounded Off-diagonal 2 × 2 Operator Matrices in Banach Spaces

Hardt, Volker; Mennicken, Reinhard

doi:10.1007/978-3-0348-8323-8_12

Cited by 8 publications

(12 citation statements)

References 10 publications

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“…Throughout this section let A and G be (possibly unbounded and/or non‐invertible) selfadjoint operators in the Hilbert space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$(\mathcal H,(\cdot \,,\cdot ))$\end{document}. Each of the statements in the following proposition follows from or is an easy consequence of 15, Remark 2.5] and 15, Theorem 1.1], see also 16, 32.…”

Section: Products Of Selfadjoint Operatorsmentioning

confidence: 99%

Spectral functions of products of selfadjoint operators

Azizov

Denisov

Philipp

2012

Mathematische Nachrichten

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Key words Product of selfadjoint operators, indefinite inner product, definitizable operator MSC (2010) 47A11, 47B50 Dedicated to our friend, the outstanding Ukrainian-American mathematician Eduard Tsekanovskiȋ on the occasion of his 75th anniversaryGiven two possibly unbounded selfadjoint operators A and G such that the resolvent sets of AG and GA are non-empty, it is shown that the operator AG has a spectral function on R with singularities if there exists a polynomial p = 0 such that the symmetric operator Gp(AG) is non-negative. This result generalizes a wellknown theorem for definitizable operators in Krein spaces.

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Section: Products Of Selfadjoint Operatorsmentioning

confidence: 99%

Spectral functions of products of selfadjoint operators

Azizov

Denisov

Philipp

2012

Mathematische Nachrichten

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“…Proof. It was proved in [3] and [4] that, under these hypotheses, ST and T S are densely defined, (ST ) * = T * S * and (T S) * = S * T * . Also, by [9], 1 dom (ST ) + ST has dense range if and only if 1 dom (T S) + T S has dense range.…”

Section: Indeed For Every Givenmentioning

confidence: 99%

“…We should also note that, in our approach (i.e., S and T are adjoint to each other), the matrices M −S,T and M S,−T are both symmetric. In some earlier work the role of these matrices has been played by some 2 × 2 symmetric offdiagonal matrices (see, for example, [3,4,6,15,18]).…”

Section: Introductionmentioning

confidence: 99%

On operators which are adjoint to each other

Popovici¹,

Sebestyén²

2014

Acta Sci. Math.

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Given two linear operators S and T acting between Hilbert spaces H and K , respectively K and H which satisfy the relationi.e., according to the classical terminology of M.H. Stone, which are adjoint to each other, we provide necessary and sufficient conditions in order to ensure the equality between the closure of S and the adjoint of T. A central role in our approach is played by the range of the operator matrixWe obtain, as consequences, several results characterizing skewadjointness, selfadjointness and essential selfadjointness. We improve, in particular, the celebrated selfadjointness criterion of J. von Neumann.

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“…The first of our main objectives is to show that (t2) implies (t1) and is accomplished in Theorem 3.1. Our main tool here is a Banach-space result from [6]. Our second aim is to prove analogues of central results of [15] assuming-instead of (t3)-that the operator T [ * ] T is only definitizable over a subset Ω of C (see Definition 3.8).…”

Section: F Philipp a C M Ran And M Wojtylak Ieotmentioning

confidence: 99%