Operator Classes

Here we briefly review a few classes of operators which are particularly relevant for the study of Gaussian measures and probability on infinite-dimensional spaces. Throughout, $H$ will be a separable real Hilbert space.

Let $A$ be a linear operator on $H$.

1. (Bounded) $L(H)$ is the class of bounded linear operators on $H$, equipped with the operator norm $|A| = \sup_{|x|\leq 1} |Ax|$.
2. (Compact) $S_{\infty}(H) \subset L(H)$ is the class of compact operators, where $A$ is compact if it maps bounded sets to precompact sets. $S_\infty(H)$ is a subspace of $L(H)$, i.e., it is a closed vector space.
3. (Hilbert-Schmidt) $S_2(H) \subset L(H)$ is the class of Hilbert-Schmidt operators, consisting of $A \in L(H)$ with finite Hilbert-Schmidt norm $|A|_2^2 = \sum_{k=1}^\infty |A e_k|^2$ for any orthonormal basis $(e_k)_{k=1}^\infty$ of $H$. The value of $|A|_2$ is independent of the choice of basis. Observe that $|A|_2^2 = \sum_{i,j=1}^\infty a_{ij}^2$, where $a_{ij} = \langle A e_i, e_j \rangle$ are the "entries" of $A$. The space $S_2(H)$ can be equipped with the inner product $\langle A, B \rangle = \sum_{i=1}^\infty \langle A e_i, B e_i \rangle$, under which $S_2(H)$ is a separable Hilbert space.

Unlike $S_\infty(H)$, the space of compact operators $L_2(H)$ is only a closed subset of $L(H)$ (in the norm topology) when $H$ is finite dimensional.

This proposition provides us with a few interpretations. First, the natural embedding $S_2(H) \hookrightarrow L(H)$ is continuous. Second, a bounded linear operator is Hilbert-Schmidt if and only if its adjoint $A^*$ is Hilbert-Schmidt. Third, $S_2(H)$ is a two-sided ideal of $L(H)$ (i.e., closed under multiplication from both sides by elements of $L(H)$).

In fact, one can say more. The operator $T$ can be given by the modulus of $A$, i.e., $T = (A^*A)^{1/2}$. Moreover, $U$ can be taken to vanish on the kernel of $T$ and $|U| \leq 1$. Recall that we can loosely think of $T$ as a scaling operator and $U$ as a rotation operator.

Nuclear Operators

Let $A \in S_\infty(H)$ be compact. Since its modulus $T = (A^* A)^{1/2}$ is compact, self-adjoint, and positive, the spectral theorem shows that there exists an orthonormal basis for $H$ consisting of eigenvectors for $H$, and that all eigenvalues are real and non-negative. These non-negative eigenvalues of $(A^* A)^{1/2}$ are called the singular values of $A$.

For $A \in S_1(H)$, its trace is given by \[ \text{tr} A = \sum_{n=1}^\infty \langle A e_n, e_n \rangle \] which converges absolutely and is independent of the choice of orthonormal basis $(e_n)_{n=1}^\infty$. We equip $S_1(H)$ with the norm $|A|_1 = \sum_{n=1}^\infty \alpha_n$, where $\alpha_n$ are again the eigenvalues of $(A^* A)^{1/2}$ counted with multiplicity. Under this norm, $S_1(H)$ is a Banach space.

In particular, the natural embeddings $S_1(H) \hookrightarrow S_2(H)$ and $S_1(H) \hookrightarrow L(H)$ are continuous. If $A \in S_2(H)$ then $A^2 \in S_1(H)$. The nuclear operators form a two-sided ideal in $L(H)$.

We have the inclusions $S_1(H) \subset S_2(H) \subset S_\infty(H) \subset L(H)$, and generally these inclusions are strict. The set $S_0(H)$ of finite-rank operators (i.e., those with finite-dimensional images) is dense in $S_1(H), S_2(H), S_\infty(H)$ but not $L(H)$.

S-Operators

For example, in $\ell^2$, the diagonal operator $A = \text{diag}(a_n)_{n=1}^\infty$ is an $S$-operator if and only if $a_n \geq 0$ and $\sum_{n=1}^\infty a_n < \infty$. In this case, $\text{tr}A = \sum_{n=1}^\infty a_n$.

Now, suppose $A$ is an $S$-operator. In particular, it is compact and self-adjoint, so by the spectral theorem there is an eigenbasis $(e_n)_{n=1}^\infty$ with $A e_n = \lambda_n e_n$ and $A x = \sum_{n=1}^\infty \lambda_n \langle x, e_n \rangle e_n$. By positivity, $\lambda_n \geq 0$. Since $A$ is self-adjoint and positive, $(A^* A)^{1/2} = A$ and in particular its singular values coincide with $(\lambda_n)$. Thus, as $A$ is nuclear, $|A|_1 = \sum_{n=1}^\infty \lambda_n < \infty$.

In fact, a converse to this holds, and we obtain a characterization of $S$ operators as those with non-negative summable eigenvalues.

We use $L_S(H)$ to denote the set of all $S$-operators. It is a convex cone in $S_1(H)$ (i.e., closed under linear combinations with non-negative coefficients). Moreover $L_S(H)$ is closed under the nuclear topology, i.e., if $(S_n) \subset L_S(H)$ and $|S_n - S|_1 \to 0$, then $S \in L_S(H)$.