There is an idea to consider instead of inner product the expression of the form x*(Tx), where x* is a norm-1 linear functional, x is a norm-1 element and x*(x) = 1. With this idea in hand one can generalize concepts of self-adjoint operators and positive operators. Of course, this generalization does not work as good as in Hilbert spaces, but nevertheless sometimes is useful. See the following book, where this idea is used to define self-adjoint and positive elements in Banach algebras.

F.F.Bonsall and J.Duncan,

Numerical Ranges of Operators on Normed Spaces and of Elements

of Normed Algebras, London Math. Soc. Lecture Note Series

2, Cambridge, 1971

Thank you prof. Kadets for your prompt replay; I will see the book as you kindly suggest. The pairing notation I cited means x*(Tx) as you write. My problem is mainly related to the duality map, which is nonlinear. This way, I do not know if >= C holds for generic linear operators T.

Sure, for a general T the inequality >= C can fail: one can give an example of a bijective linear operator in 2-dimensional \ell_\infty, such that for some a, b the left hand side is 0, but the right hand side is strictly positive.

Concerning the description of those T that satisfy >= C I have no idea.

In a normed space X, we define a monotone operator A : X --> X* if satisfies :

< Ax - Ay, x - y > >= 0 for all x, y in D(A)

where means the action of Ax into y. Now, if A is linear, monotonicity becomes :

>= 0 for all x in D(A).

By the way, what would be the significance of this:

>= C.

In the case of a Hilbert space, J = I, the identity operator and hence T would be an isometry, but outside of Hilbert spaces, what would be J(Ta)?, which, by the way, J is a multivalued operator for general normed spaces.

Thank you for your contribution. To simplify the problem, we can just consider L^p Banach spaces, 10 real constant. If T is the identity, it holds with C=1, but T=I is not a useful case... Anyway, thanks to continuity arguments, a class of operators (larger than the sole identity...) could exist. If the Banach space is finite-space Hilbert space (so that J=I), the inequality holds for any T such that T^*T has trivial kernel (i.e., Ker(T)=0), but unfortunately I do not work in Hilbert space, since I need to work in Banach space.

Claudio,

In your case, J is single-valued, as you said it. So you are trying to identify operators T that fulfill the iquality, Am I reading you correctly?

For a linear operator T, the inequality x*(Tx) >= c||x||= cx*x, (where x* is a tangent functional) is equivalent to the estimate || (λ+Τ)x || >= Re(λ+c) ||x|| (whatever the sign of c). The latter is a resolvent estimate if (λ+Τ) is of a dense range.

A non-linear analog is (x-y)*(Tx - Ty) >= c || x-y || = c (x-y)*(x-y). It is equivalent to the estimate || (λ+Τ)x - (λ+Τ)y || .=< Re(λ+c) || x-y || etc.