Generalized Resolvents and Spectral Functions of a Matrix Generalization of the KREIN-FELLER Second Order Derivative

✍ Scribed by L. P. Klotz; H. Langer

• Publisher: John Wiley and Sons
• Year: 1981
• Tongue: English
• Weight: 1012 KB
• Volume: 100
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19811000109

✦ Synopsis

If nt is a nondecreasing function on the interval [O, I], 0 -=I -= 00, in [3] a descrip-(12 I ] lilotz/Langer, Generalized Resolvents and Spectral Functions and the CAUCHY-SCHWARZ inequality implies t (Os-tss). The il;tegral equation for @(.; z,,) gives 0 -

The proof of (ii) is similar.

2. Corresponding canonieal systems

2.1. By C(. ; z ) we denote the 29 X Xq-matrix function

It is easy to check that V ( . ; z ) satisfies the integral equation (2.1) X

U ( r ; z ) = 1 2 , -J J (ZdH,(s) U ( s ; z ) + d H t ( s ) V(s; 2)). zE[O, I ] ,

0which can be written as

J d U ( x ; z ) = w / H O ( x ) U ( x ; z ) + d H l ( z ) U ( z ; z), z€[O, I ] , U ( 0 -; z ) = 1 2 * .

Here

The matrix functions U ( -; z ) satisfy the fundamental identity z (2.2) L~(s; i)* JC'(S; z ) -J = ( z -: ) J U ( S ; [)* dHo(s) U(5; Z) 0 -( 2 , ;E&. s E [ O , I ] ) . Indeed, we have d ( U ( s ; ()* J C ( s ; z ) ) = d U ( s ; <)* J V ( s : z ) + U ( s ; [)* JdL7(s; x ) = -4"(s; 5')* f Z H 0 ( S ) U ( s ; z ) + z L ; ( s ; [)* dH,(s) U ( s ; 2) , and (2.2) follows by integration from 0 -t o x. The relation (2.2) iitiplies that the kernel K,(z;[):=(z-~)-'(~(z;[)* J U ( X ; Z ) -J )