A structure theorem on non-homogeneous linear equations in Hilbert spaces

Abstract: A very particular by-product of the result announced in the title reads as follows: Let

(X, < c d o t, c d o t >)

be a real Hilbert space,

T : X o X

a compact and symmetric linear operator, and

z i n X

such that the equation

T (x) - | T | x = z

has no solution in

X

. For each

r > 0

, set

g a m m a (r) = s u p_{x i n S_{r}} J (x)

, where

J (x) = < T (x) - 2 z, x >

and

S_{r} = x i n X : | x |^{2} = r

. Then, the function

g a m m a

is

C^{1}

, increasing and strictly concave in

] 0, + i n f t y [

, with

g a m m a^{'} (] 0, + i n f t y [) =] | T |, + i n f t y [

; moreover, for each

r > 0

, the problem of maximizing

J

over

S_{r}

is well-posed, and one has

T(hat x_r)-gamma'(r)hat x_r=z

where

h a t x_{r}

is the only global maximum of

J_{| S_{r}}

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