A Note on Riesz Bases of Eigenvectors of Certain Holomorphic Operator-Functions

✍ Scribed by Joseph Lutgen

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 135 KB
• Volume: 255
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.2000.7154

✦ Synopsis

Operator-valued functions of the form

compact-valued and holomorphic on certain domains ⍀ ; ‫ރ‬ are considered in separable Hilbert space. Assuming that the resolvent of A is compact, its eigenvalues are simple and the corresponding eigenvectors form a Riesz basis for H H of finite defect, it is shown that under certain growth conditions 5 Ž .Ž . y1 5 on Q A y the eigenvectors of A A corresponding to a part of its spectrum also form a Riesz basis of finite defect. Applications are given to operator-valued Ž . Ž . y1 functions of the form A A s A y q B y D C and to spectral problems in 2 Ž . Ž . Ž . Ž . Ž . Ž . Ž . L 0, 1 of the form yf Љ x q p x, f Ј x q q x, f x s f x with, for example, Dirichlet boundary conditions.