Unitary Operator

Abstract

Unitary operator, a sharpening of the concept of an isometric operator. A linear operator J defined on a complex (real) Banach space X (Hilbert space) with values in some complex (real) Banach space Y is called isometric or an isometry if it preserves the norm, i.e., J φ = =φ for all φ ∈ X. An isometric operator is bounded (operator) with norm J = 1, invertible, and the range R J is a closed (Hilbert space) submanifold of Y which is, even in the case Y = X , in general smaller than Y (if X and Y have the same finite dimension, then R J = Y). The inverse operator J −1 is an isometry with domain D J −1 = R J and the range R J −1 = X. Two Banach spaces X and Y are called (norm-) isomorphic if there exists an isometry from X to Y such that R J = Y. An isometric operator J defined on a complex (real) Hilbert space H with values in some complex (real) Hilbert space K automatically preserves the scalar products also, i.e., J φ|J ψ = =φ|ψ for φ, ψ ∈ H. Such an operator is called unitary [1-6] if H and K are complex Hilbert spaces and if its range is K. That is, a linear operator U from some complex Hilbert space H to some other complex Hilbert space K is unitary if (i) D U = H, (ii) Uφ|Uψ = =φ|ψ for φ, ψ ∈ H, and (iii) R U = K. The inverse U −1 is also unitary where, in the case of H = K, U −1 = U * holds (the assumption H = K is not necessary, but corresponds to the definition of the adjoint operator given in the section operator). The following example shows that an isometric operator acting in a complex Hilbert space is in general not unitary. Let φ 1 , φ 2 ,. .. be a complete orthonormal system of an infinite-dimensional separable Hilbert space H. For every vector ψ ∈ H, ψ = ∞ i=1 α i φ i , ∞ i=1 |α i | 2