ON NUMERICAL RANGE AND NUMERICAL RADIUS OF A SPECIAL PAIR OPERATOR MATRICES

Abstract We introduce concepts of essential numerical range for the linear operator pencil $\lambda \mapsto A-\lambda B$. In contrast to the operator essential numerical range, the pencil essential numerical ranges are, in general, neither convex nor even connected. The new concepts allow us to describe the set of spectral pollution when approximating the operator pencil by projection and truncation methods. Moreover, by transforming the operator eigenvalue problem $Tx=\lambda x$ into the pencil problem $BTx=\lambda Bx$ for suitable choices of $B$, we can obtain nonconvex spectral enclosures for $T$ and, in the study of truncation and projection methods, confine spectral pollution to smaller sets than with hitherto known concepts. We apply the results to various block operator matrices. In particular, Theorem 4.12 presents substantial improvements over previously known results for Dirac operators while Theorem 4.5 excludes spectral pollution for a class of nonselfadjoint Schrödinger operators which has not been possible to treat with existing methods.

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On 𝔸-numerical radius inequalities for 2 × 2 operator matrices

Linear and Multilinear Algebra ◽

10.1080/03081087.2020.1810201 ◽

2020 ◽

pp. 1-21

Author(s):

Nirmal Chandra Rout ◽

Satyajit Sahoo ◽

Debasisha Mishra

Keyword(s):

Operator Matrices ◽

Numerical Radius

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Polynomials in a hermitian element

Glasgow Mathematical Journal ◽

10.1017/s0017089500007187 ◽

1988 ◽

Vol 30 (2) ◽

pp. 171-176 ◽

Cited By ~ 2

Author(s):

M. J. Crabb ◽

C. M. McGregor

Keyword(s):

Banach Algebra ◽

Dual Space ◽

Numerical Range ◽

Numerical Radius ◽

Hermitian Element ◽

Unital Banach Algebra

For an element a of a unital Banach algebra A with dual space A′, we define the numerical range V(a) = {f(a):f ∊ A′, ∥f∥ = f(1) = 1}, and the numerical radius v(a) = sup{⃒z⃒:z ∊ V(a)}. An element a is said to be Hermitian if V(a) ⊆ ℝ ,equivalently ∥exp (ita)∥ = 1(t ∊ ℝ). Under the condition V(h) ⊆ [-1, 1], any polynomial in h attains its greatest norm in the algebra Ea[-1,1], generated by an element h with V(h) = [-1, 1].

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On a Numerical Radius Preserving Onto Isometry onL(X)

Journal of Function Spaces ◽

10.1155/2016/9183135 ◽

2016 ◽

Vol 2016 ◽

pp. 1-3

Author(s):

Sun Kwang Kim

Keyword(s):

Banach Space ◽

Hilbert Space ◽

Numerical Range ◽

Complex Banach Space ◽

Uniform Convexity ◽

Numerical Radius ◽

Uniform Smoothness

We study a numerical radius preserving onto isometry onL(X). As a main result, whenXis a complex Banach space having both uniform smoothness and uniform convexity, we show that an onto isometryTonL(X)is numerical radius preserving if and only if there exists a scalarcTof modulus 1 such thatcTTis numerical range preserving. The examples of such spaces are Hilbert space andLpspaces for1<p<∞.

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