T ∗ T always has a positive selfadjoint extension

Sebestyén, Zoltán; Tarcsay, Zsigmond

doi:10.1007/s10474-011-0154-7

Cited by 25 publications

(13 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As it was shown in [13], T * T is not necessarily selfadjoint but it always has a positive selfadjoint extension, namely its smallest, so called Krein-von Neumann extension. Below we give an alternative proof of that statement, constructing a selfadjoint extension of T by means of the canonical graph projection of H × K onto G(T ).…”

Section: Positive Selfadjoint Operatorsmentioning

confidence: 96%

See 1 more Smart Citation

On the adjoint of Hilbert space operators

Sebestyén

Tarcsay

2018

Linear and Multilinear Algebra

Self Cite

View full text Add to dashboard Cite

In general, it is a non trivial task to determine the adjoint S * of an unbounded operator S acting between two Hilbert spaces. We provide necessary and sufficient conditions for a given operator T to be identical with S * . In our considerations, a central role is played by the operator matrix M S,T = I −T S I . Our approach has several consequences such as characterizations of closed, normal, skew-and selfadjoint, unitary and orthogonal projection operators in real or complex Hilbert spaces. We also give a self-contained proof of the fact that T * T always has a positive selfadjoint extension.1991 Mathematics Subject Classification. Primary 47A05, 47B25. Key words and phrases. Adjoint, closed operator, selfadjoint operator, positive operator, symmetric operator.

show abstract

Section: Positive Selfadjoint Operatorsmentioning

confidence: 96%

“…Instead of using the defect index theory developed by J. von Neumann, our method involves the range of M S,S . In Theorem 7.3 we also present a new proof of the fact that T * T always has a positive selfadjoint extension (see [13]). In section 8 we characterize densely defined closed operators.…”

Section: Introductionmentioning

confidence: 98%

On the adjoint of Hilbert space operators

Sebestyén

Tarcsay

2018

Linear and Multilinear Algebra

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the case that S is densely defined Theorem 2.4 gives the following result. in [24] these factorizations for S ≥ 0 were constructed in another way. Theorem 2.4 involves the Friedrichs extension S F of S. There is a similar result for the Kreȋn extension S K of S. The Kreȋn extension in the nonnegative case was introduced and studied in [19].…”

Section: Thenmentioning

confidence: 99%

Factorized sectorial relations, their maximal-sectorial extensions, and form sums

Hassi¹,

Sandovici²,

Snoo

2019

Banach J. Math. Anal.

View full text Add to dashboard Cite

In this paper sectorial operators, or more generally, sectorial relations and their maximal sectorial extensions in a Hilbert space H are considered. The particular interest is in sectorial relations S, which can be expressed in the factorized formwhere B is a bounded selfadjoint operator in a Hilbert space K and T : H → K or T : K → H, respectively, is a linear operator or a linear relation which is not assumed to be closed. Using the specific factorized form of S, a description of all the maximal sectorial extensions of S is given with a straightforward construction of the extreme extensions S F , the Friedrichs extension, and S K , the Kreȋn extension of S, which uses the above factorized form of S. As an application of this construction the form sum of maximal sectorial extensions of two sectorial relations is treated.2010 Mathematics Subject Classification. Primary 47B44; Secondary 47A06, 47A07, 47B65.

show abstract

“…Note that if T is not closed, then T * (I + iB)T is a sectorial relation which may have maximal sectorial extensions, such as T * (I + iB)T * * and some of these extensions have been determined in [5]; cf. [10].…”

Section: Introductionmentioning

confidence: 99%

A class of sectorial relations and the associated closed forms

Hassi¹,

Snoo²

2019

Preprint

View full text Add to dashboard Cite

Let T be a closed linear relation from a Hilbert space H to a Hilbert space K and let B ∈ B(K) be selfadjoint. It will be shown that the relation T * (I + iB)T is maximal sectorial via a matrix decomposition of B with respect to the orthogonal decomposition H = dom T * ⊕ mul T . This leads to an explicit expression of the corresponding closed sectorial form. These results include the case where mul T is invariant under B. The more general description makes it possible to give an expression for the extremal maximal sectorial extensions of the sum of sectorial relations. In particular, one can characterize when the form sum extension is extremal.

show abstract

T ∗ T always has a positive selfadjoint extension

Cited by 25 publications

References 9 publications

On the adjoint of Hilbert space operators

On the adjoint of Hilbert space operators

Factorized sectorial relations, their maximal-sectorial extensions, and form sums

A class of sectorial relations and the associated closed forms

Contact Info

Product

Resources

About