Spectral Analysis of the Bounded Linear Operator in the Reproducing Kernel Space<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>D</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>

Guo, Lihua; Li, Songsong; Wu, Boying; Zhang, Dazhi

doi:10.1155/2014/691461

Search citation statements

Order By: Relevance

Paper Sections

Select...

Proposition 7 Let X Be a Complex Banach Space And T ∈ B(x) T...1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2022

Publication Types

Select...

Article1

Relationship

Self Cite0

Independent1

Authors

Journals

Cited by 1 publication

(1 citation statement)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…is closed and bounded implies Π(T ) is a compact subset of C(In Euclidean metric spaces R n , C n compactness implies closedness and boundedness ( [5], [7] and [8])).…”

Section: Proposition 7 Let X Be a Complex Banach Space And T ∈ B(x) T...mentioning

confidence: 99%

Spectral Radius of a Normal Operator

Simiyu

Alosa

Fanuel

2022

ARJOM

View full text Add to dashboard Cite

Let X be a Complex Banach space and T be a bounded operator in X. The number sup {|λ| : λ ∈ σ(T)} (where σ(T) is the spectrum of T and σ(T) ̸= ϕ) is called the spectral radius of T and denoted by r(T). Since λ ≤ ∥T∥for all λ ∈ σ(T), it follows that r(T) ≤ ∥T∥. The spectral mapping theorem implies that r(Tn) = (r(T))n for every positive integer n. It frequently turns out that it is easy to compute the spectral radius of an operator even if it is hard to _nd the spectrum. This is often made easy by the spectral radius formula. Let H be a Hilbert space and T be a bounded linear operator in H. In this paper we show that if T is normal, then Tn is normal for each n ∈ N and ∥Tn∥ = ∥T∥n. Consequently, we use the spectral radius formula to show that r(T) = ∥T∥. Moreover, we show that if X is a Complex Banach space and T is bounded in X then there is a λ belonging to the spectrum of T such that |λ| = r(T). Let H be a Complex Hilbert space and T be a bounded operator in H which is normal; we show that ∥T∥ = sup {|Tx, x| : x ∈ H and ∥x∥ = 1} and the residual spectrum of T is void.

show abstract

“…is closed and bounded implies Π(T ) is a compact subset of C(In Euclidean metric spaces R n , C n compactness implies closedness and boundedness ( [5], [7] and [8])).…”

Section: Proposition 7 Let X Be a Complex Banach Space And T ∈ B(x) T...mentioning

confidence: 99%