The Stability Radius of Fredholm Linear Pencils

✍ Scribed by C. Badea; M. Mbekhta

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 116 KB
• Volume: 260
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.2000.7445

✦ Synopsis

Let T and S be two bounded linear operators from Banach spaces X into Y, Ž . and suppose that T is Fredholm and dim N T y S is constant in a neighborhood Ž . Ž . of s 0. Let d T; S be the supremum of all r ) 0 such that dim N T y S and Ž . < < codim R T y S are constant for all withr. It is a consequence of more Ž . general results due to H. Bart and D. C. Lay 1980, Studia Math. 66, 307᎐320 that Ž . Ž . 1r n Ž . Ž . d T; S s lim ␥ T ; S , where ␥ T ; S are some non-negative extended n ª ϱ n n Ž . real numbers. For X s Y and S s I, the identity operator, we have ␥ T ; S s n Ž n . ␥ T , where ␥ is the reduced minimum modulus. A different representation of Ž . the stability radius d T; S is obtained here in terms of the spectral radii of generalized inverses of T. The existence of generalized resolvents for Fredholm linear pencils is also considered.