Symplectic Hilbert Spaces

Real Symplectic Hilbert Spaces

In the finite-dimensional setting, a vector space admits a symplectic structure only if its dimension is even. For a separable Hilbert space, dimension can no longer be the relevant criterion. Instead, one constructs a symplectic Hilbert space from a compatible complex structure. This section draws heavily on Sec. 2, Ner90. However, Neretin’s definition of a symplectic Hilbert space is not entirely clear, so we streamline it here:

Symplectic Hilbert Space

A symplectic Hilbert space is a real separable Hilbert space with a compatible complex structure . That is, a bounded linear operator such that

where is the adjoint of with respect to the inner product .
Then we can define as the canonical symplectic form on it.

Remark

is the exact condition that makes alternating:

Restricted Symplectic Group

Suppose is a symplectic Hilbert space, then we write for the restricted symplectic group, that is, a subgroup of such that the operators can be written as for some -unitary operator and some Hilbert-Schmidt operator on .

Remark

captures the idea of “almost unitary” symplectic operators on a real Hilbert space. Note that if is finite-dimensional, then coincides with . To determine whether a symplectic operator on belongs to , it suffices to check that is a Hilbert—Schmidt perturbation of the identity. Indeed, if , then is necessarily self-adjoint and compact. By the spectral calculus, is again a Hilbert—Schmidt perturbation of the identity. Hence, by the polar decomposition,

for some partial isometry . Since is invertible, this partial isometry is in fact unitary.
Moreover, note that both and
are Hilbert—Schmidt perturbations of the identity. Therefore, is a subgroup of .

For a concrete example of an element of , see example below, after the introduction of the complexified symplectic Hilbert space.

Complexified Symplectic Hilbert Spaces

In fact, real symplectic Hilbert space is not well suited to our purposes, as it lacks the necessary structure and does not naturally fit into the framework of Fock space representations. We therefore pass to its complexification. Indeed, in Neretin’s paper, a symplectic Hilbert space is constructed directly in this complexified sense. The complexified symplectic Hilbert space carries additional structure, notably a polarisation presented in the proposition, which allows each vector to be decomposed into a creation-like part and an annihilation-like part. We will discuss this at the beginning of the next section.
Let us first examine these additional structures, apart from the polarisation. Let be a symplectic Hilbert space with the canonical symplectic form . Define to be the complexification as a Hilbert space. Then the following additional
structures appear canonically on :

• A real structure (i.e., conjugation) , that flips the sign of the imaginary part;
• An inner product , extending sesquilinearly: ;
• A skew Hermitian form extends sesquilinearly: ;
• The symplectic form extends to bilinearly: ;
• The complex structure extends to linearly: .

Polarisation of Complexified Symplectic Hilbert Spaces

Suppose is a symplectic Hilbert space with complexification . Let , then orthogonally, and are Lagrangian.

Proof Let us first show that . For any , we can write and
, then clearly this forms an orthogonal decomposition with . Moreover, the only vector that can lie in both and is the zero vector, so the decomposition is unique. In fact, this is the eigenspace decomposition of the operator .
For , we have . For all , there holds so and is isotropic.
Similarly is also isotropic. To show is maximal, it suffices to show that is nondegenerate on for . Suppose for all , then will vanish on the whole . Since is nondegenerate, , thus is nondegenerate on .

Remark

The data of a symplectic Hilbert space is equivalent to that of a polarised complex Hilbert space equipped with a compatible real structure. Indeed, given a tuple

consisting of a complex Hilbert space with inner product , a real structure , and a closed subspace such that and orthogonally, we can recover:

1. A real Hilbert subspace ;
2. A bounded linear map for that forms a complex structure on .

So that is a symplectic Hilbert space.

Henceforth in this note, unless otherwise specified, denotes a complexified symplectic Hilbert space consisting of the following data:

e.g. Utilising the polarisation on complexified symplectic Hilbert space, we can easily construct a nontrivial example of an operator in . Let be an orthonormal basis of , so is an orthonormal basis of . Fix some , define by and let act as the identity on all other basis vectors. Note that is a basis of . Moreover, Hence . Since has finite rank, it is Hilbert-Schmidt, thus is Hilbert-Schmidt as well. Furthermore, is diagonal, and its diagonal entries occur in reciprocal pairs. In particular, Therefore is symplectic, and hence belongs to .

We now introduce a semigroup on a complexified symplectic Hilbert space, which serves as an extension of . Before doing so, we recall the usual notion of transpose:

Transpose

Suppose , then is defined as the unique element in such that
Moreover, for an operator , the transpose of is defined as
where is the adjoint of w.r.t the inner product .

Correct Operator & Potapov–Ginzburg Matrix

Given a complexified symplectic Hilbert space , suppose is an (possibly unbounded) operator on . If there exists

satisfying

where

1. ;
2. ;
3. and are Hilbert-Schmidt,

then is called a correct operator, and is called the Potapov—Ginzburg transformation (matrix) of . The collection of correct operators forms a semigroup under the usual composition of operators. Additionally, we endow with the strong Hilbert-Schmidt topology such that if and only if and strongly.

Remark

Note that the bottom left entry is , not . Indeed, it must be . To see this, introduce the notation and to distinguish the domain and codomain of (although they coincide as Hilbert spaces, we distinguish them notationally). Then so it is natural that whereas

Moreover, can be described as follows. Let

be the conjugation isomorphisms. Then is the adjoint of with respect to the inner product.

We can use the Potapov-Ginzburg matrix to describe these operators in . Given an operator with the block matrix presentation with respect to the decomposition , we have If is invertible, then we can rearrange the above equation to get Conversely, if with invertible , we can express the block matrix of (w.r.t the decomposition ) using , and :

e.g. Recall the operator from the previous example, we claim it is correct. It has the block form where and . If the Potapov-Ginzburg transformation exists, then by the above equations, it is given by It is straightforward to verify that Since has finite rank, both and are Hilbert-Schmidt and have operator norm Hence is correct. So we have obtained an example of an operator which is both restricted symplectic and correct.

as a subgroup of

More generally, embeds into as a subgroup via complexification, and appears as a boundary of . This is established in the following theorem, adapted from Sec. 2.3, Ner90:

Theorem

Suppose is a symplectic Hilbert space with complexification . For any almost unitary symplectic operator , it can be complexified to an operator on . Then

Proof

: Suppose , that is , We can express the block matrix of using , and : To show that , it is sufficient to show that is a Hilbert-Schmidt perturbation of the identity. By , we can simplify to Note that is Hilbert-Schmidt, so all of , and are Hilbert-Schmidt. Additionally, for the right-bottom entry, we have Now use the resolvent identity and the fact that is a trace-class, so the right-bottom entry is Hilbert-Schmidt as well.

References

Neretin, Holomorphic extensions of representations of the group of diffeomorphisms of the circle, 1990