A unitary operator preserves the inner product, doesn't it?

Actually, the right concept is unitarily equivalent representations. The physical reason for this is, in intuitive terms, that nothing physically relevant is lost. Note, however, that if it is only an isometric isomorphism you can always make it unitary by projecting onto an appropriate subspace of the Hilbert space. Ricardo Weder.    

First, a C-linear isometric isomorphism is automatically unitary. This is because an isomorphism is invertible and = for all x, so U*U = 1. However since U is invertible the left inverse U* is also a right inverse, from which it follows that UU* = 1 

Second, Unitary equivalence is the natural equivalence for C* algebras because it means that the representations are equivalent as *-representations. 

To see this, consider a representation of a C* algebra A as a *-homomorphism

f: A --> B(H)

for some Hilbertspace H. One can now consider two versions of unitary equivalence of representations f:A --> B(H1) and g:A--> B(H2): internal and external.  The representations are internally equivalent if H1 = H2 and there is an u \in A with u*u = uu* =1 and g(a) = f(uau*)= f(u)f(a)f(u)*. Note that f(u) is unitary and

g(a*) = f(ua*u*) = f( (uau*)*) = f(uau*)*

as it should for a *-representation.  The *-representations are externally unitarily equivalent if there is a unitary U:H1 -->H2 and g(a) = Uf(a)U*. Note that internally equivalent  is stronger than externally equivalent and that

g(a*) = Uf(a*)U* = Uf(a)*U* = (Uf(a)U*)*

as it should. AFAIK internal equivalence is actually the more interesting equivalence but I could be wrong on that.. 

A bijective linear map U:H -> K between two Hilbert spaces is called unitary if preserves the scalar product, hence the norm. In particular, by polarization, a surjective isometry is a unitary operator.

Two representations p on H and q on K of a C*-algebra A are called unitarily equivalent if there exists a unitary operator U:H -> K such that q(a)U=Up(a), for any a in A.

This is exactly what the physicists (and the mathematical physicists) mean when they request unitary equivalence of representations.

Let me add that there exist weaker forms of equivalence between representations that are useful in physics, e.g. weak equivalence.

If one has different Hilbert spaces, one would have the inner product defined separately 

on each of them. So preserving THE inner product may not be the correct way to put it.

There are two inner products around on two different vectors spaces as such. That was all that I had on mind.

the sentence "preserves the inner product" was a shortcut for

(Ux,Uy)_K=(x,y)_H

One also talks about a unitary map between two different Hilbert spaces. A unitary map is an isometric isomorphism. Think of the position and momentum representations of quantum mechanics. The Fourier transform maps the Hilbert space of the position space realization to the Hilbert space of momentum space realization. Think also of the so-called Fock space realization of the harmonic oscillator problem. Once again there is a unitary transformation which maps the Schroedinger representation to the Fock representation. The point is that the scalar products should be preserved under the map.

My query is perhaps simply a matter of terminology. But it does look important that one notes that inner products could be different. 

I am trying to understand Araki's work on quasi equivalence. There is a single vector space with two inner products on it. Each comes from a quasifree state. The symplectic structure on that space gets related to two linear operators called polarizators, each because of an inner product. Further, Araki showed that their square roots differing by a Hilbert Schmidt operator is equivalent to the quasifree states being quasi equivalent. I was trying to see if I could construct an operator which was "unitary" so that I could work more easily on this. So this question of terminology arose in my mind.

The idea of wondering about a "unitary" operator was an attempt to see the fact that quasi free states may generate representations which are unitarily equivalent, but not always!

A single vector space with two different inner products counts as two  different Hilbert spaces. A  Hibertspace is after all a pair of a complete real or complex vector space together with a sesquilinear innerproduct.

Actually, I should be putting a separate question, but I think it looks an issue of minor difference in presentation styles.

I am trying to follow Araki's work as well as that of R.Verch (Local definiteness, primarity and quasiequivalence of quasifree Hadamard quantum states....) Araki's work begins with real Hilbert spaces while Verch develops the formalism emphasizing complexified "solution" spaces. The idea of "polarizations" has been concisely introduced on such complexified spaces in the latter. The way the polarization is presented in Lemma 3.4 looks like "real" plus i times "imaginary" parts. But that impression is not correct. The "real" part is actually a complexified inner product on a complexified space and so is

 the "imaginary" part which is a complexified symplectic product. There are two such complexified inner products around and on the same solution space.  Now if I have an operator which preserves "the" inner product, it would be really nice to use it further an try to understand Araki's proof. I suppose it would be then important to ask when such an operator could be non-invertible. For we are going beyond unitarity in general when we are looking for quasiequivalence. 

Araki's work is presented in a different manner as three separate results leading to necessary and sufficient conditions. I am trying to follow it, but would certainly welcome any pointers. Thanks in anticipation.

I would like to ask if any paper has presented a concise version of Araki and Yamagami's proof. I am finding it quite tedious to see through the constructions that they have used.

I start from a complexified symplectic vector space. Then I complexify the symplectic product. Then it becomes quite opaque to me. The proof involves a "real subspace"

From the point of view of the original space, this subset is no longer a subspace (over the

original field which is the set of complex numbers)  There must be some clean way to

define "real subspaces". Thanks in anticipation.