By B. D. Sleeman
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Extra info for Multiparameter Spectral Theory in Hilbert Space (Research Notes in Mathematics Series)
Ak) • D. 4: LeDIII4 D .. > j•l J st1J .. r J. f That D n n c j=l - and for all f e D. o, i = 1, •.. , n. ) is obvious. ), then j•l J t n t -1 n t r S .. r. f • - r S .. A0 r Ak AOk. f j•l 1l J j=l 1J k•l J n .. •• , n, and so -- st1J .. ·.. (B)f, ~n 1 1 i • 1, ••• , n. ) • D and J s"f. r. f 1J J This proves the lemma. • o, i - 1, ••• , n. ) ==>lim r. n J J exists and equals r. 1: The operators r.. (B)exists}. ), (ii) J n I r. (B)f. ) • D and for J D. n I s!. f j•l 1J J • o, i . 1, •• • 1 D. 2 , to arrive at a spectral theory and associated Parseval equality.
B)g. f,gJ. B-tlin 1 B-tlin 1 1 * Thus g E D(r. , j • 1, ••• , n has deficiency. g J = ig, 11< r. g ... -ih J ( i . -1-1) implies that g. f,g] • [f,r. f,g) • (f,A0 r. g) J J for all f E D for all f E D. (B)W(B)f. (B)W(B)f,g) • (W(B)f,A0 r. , for all f E H, (f,Wb)Aj(B)g) • (f,W(B)AO r j g) for all f E H that is 44 o. 3) Now define operators P and Q by (A1 - il)t •••••• (An - il)t: D+H . -lt •••• (A - il) -lt : H + D. Q- (Al - 11) n p- Clearly Q is bounded and for all f QP E H we have PQ • I while for all g = I.
The equations n -A0 A. f. + 1 1 r AJ. J. f j=l 1 1• - o • i • 1, ••• , n, with n r j-o a. A. • 1, has a non-trivial solution. J J Corollary 3. 3: is compact. ~: Suppose each of the operators A. : H. : a provided a 0A0 • 1. The theory described in this chapter is a generalisation of some work of Browne  and includes that of Klllstr8m and Sleeman . References 1 F. V. Atkinson, Multiparameter eigenvalues problems Vol 1, Matrices and compact operators. Academic Press, New York (1972). J. Browne, Multiparameter spect~al theory.
Multiparameter Spectral Theory in Hilbert Space (Research Notes in Mathematics Series) by B. D. Sleeman