TY - JOUR

T1 - Repeated eigenvectors and the numerical range of self-adjoint quadratic operator polynomials

AU - Lancaster, Peter

AU - Markus, Alexander S.

AU - Psarrakos, Panayiotis

PY - 2002/12/3

Y1 - 2002/12/3

N2 - Let L(λ) be a self-adjoint quadratic operator polynomial on a Hilbert space with numerical range W(L). The main concern of this paper is with properties of eigenvalues on ∂W(L). The investigation requires a careful discussion of repeated eigenvectors of more general operator polynomials. It is shown that, in the self-adjoint quadratic case, non-real eigenvalues on ∂W(L) are semisimple and (in a sense to be defined) they are normal. Also, for any eigenvalue at a point on ∂W(L) where an external cone property is satisfied, the partial multiplicities cannot exceed two.

AB - Let L(λ) be a self-adjoint quadratic operator polynomial on a Hilbert space with numerical range W(L). The main concern of this paper is with properties of eigenvalues on ∂W(L). The investigation requires a careful discussion of repeated eigenvectors of more general operator polynomials. It is shown that, in the self-adjoint quadratic case, non-real eigenvalues on ∂W(L) are semisimple and (in a sense to be defined) they are normal. Also, for any eigenvalue at a point on ∂W(L) where an external cone property is satisfied, the partial multiplicities cannot exceed two.

UR - http://www.scopus.com/inward/record.url?scp=0036436087&partnerID=8YFLogxK

U2 - 10.1007/BF01217534

DO - 10.1007/BF01217534

M3 - Article

AN - SCOPUS:0036436087

VL - 44

SP - 243

EP - 253

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

SN - 0378-620X

IS - 2

ER -