Invariant subspaces of certain linear operators
Ky Fan · Bulletin of the American Mathematical Society · 1963
Following the investigations of Pontrj agin [6] and Iohvidov [3]on linear operators in a Hilbert space with an indefinite inner product, M. G. Kreïn [5] proved the following theorem.THEOREM (PONTRJAGIN-IOHVIDOV-KREIN).Let E be the Hilbert space of infinite complex sequences x={xi\ with convergent 2^11 |#*| 2 , with norm ||x|| = (]C£»i I x *l 2 ) 1/2 -Let n be a positive integer and let Jn(x) = Ê |*<|»-E |*,|* for x-{xi} G-E. If a linear transformation $: E-+E is continuous in the norm topology, and if( 1 ) J n (x)*zO implies J n (<£(x))^J n (x), then there exists an n-dimensional linear subspace F of E such that: (i) <j)(F)(ZF' t (ii) J n (x) ^Ofor xÇzF\ (iii) every eigenvalue of the restriction of <fi on F is of absolute value ^ 1.