Some eigenvalue inequalities

Let $\sH$ be a separable (infinite-dimensional) Hilbert space, and denote by $\sL(\sH)$ the algebra of bounded operators on $\sH$. Let $\sK = \sK(\sH)$ be the ideal of compact operators on $\sH$. Each $T \in \sK$ has a polar decomposition $T = U\vert T\vert$, where $\vert T\vert = (T^*T)^{1/2} \in \sK$, and $U \in \sL(\sH)$ is a partial isometry. This factorization is unique if we require that $U = 0$ on $\ker\vert T\vert$, since $U$ must map the range of $\vert T\vert$ isometrically onto the range of $T$.

The spectral theorem yields an orthonormal family $\{\psi_k\}$ in $\sH$, such that

$$\vert T\vert = \sum_{k\geq 0} s_k(T) \ketbra{\psi_k}{\psi_k}, \qquad T = \sum_{k\geq 0} s_k(T) \ketbra{U\psi_k}{\psi_k}.$$

(If $\vert T\vert$ is invertible, this is an orthonormal basis for $\sH$. Otherwise, we can adjoin an orthonormal basis for $\ker\vert T\vert$ to the family $\{\psi_k\}$.) Since $\phi_k := U\psi_k$ gives another orthonormal family, any $T \in \sK$ has an expansion of the form

$$T = \sum_{k\geq 0} s_k(T) \ketbra{\phi_k}{\psi_k},$$ (16)

for some pair of orthonormal families $\{\phi_k\}$, $\{\psi_k\}$.

If $V_1$, $V_2$ are unitary operators on $\sH$, we can then write

$$V_1 T V_2 = \sum_{k\geq 0} s_k(T) \ketbra{V_1\phi_k}{V_2^*\psi_k},$$

and conclude that $s_k(V_1 T V_2) = s_k(T)$ for each $k$, and hence that

$$\sg_N(V_1 T V_2) = \sg_N(T).$$

Therefore, any norm $\snorm{T}$ that is built from the sequence $\set{s_k(T) : k \in \bN}$ is unitarily invariant, that is, $\snorm{V_1 T V_2} = \snorm{T}$ for $V_1$, $V_2$ unitary.

Soon, we shall introduce a ``Dixmier trace class'' $\sL^{1+}(\sH)$, with yet another norm built from singular values, such that $\sL^1 \subset \sL^{1+} \subset \sL^p$ for $p > 1$.

Much is known about the singular values of compact operators. For instance, the following relation holds, for $T \in \sK$:

$$s_k(T) = \inf\set{ \Vert T(1 - P)\Vert : P = P^2 = P^*, \dim P(\sH) \leq k}.$$ (17)

This comes from a well-known minimax principle: see [RS], for instance. The infimum is indeed attained at the projector $Q$ of rank $k$ whose range is $Q(\sH) := \spn\{\psi_0,\dots,\psi_{k-1}\}$, when $T$ is given by (), since $T(1 - Q) = T - TQ = \sum_{j\geq k} s_j(T) \ketbra{\phi_j}{\psi_j}$ is an operator with norm $\Vert T - TQ\Vert = s_k(T)$.

The triangle inequality in Corollary  is not good enough for our needs: our goal is get an additive functional, rather than just a subadditive one. The next step is to extract from () a sort of ``wrong-way triangle inequality'', at least for positive compact operators.

We see that the functional $A \mapsto \sg_N(A)/\log N$ is not far from being additive functional on the positive cone $\sK_+$. But to get a truly additive functional, we must try to take the limit $N \to \infty$, and here things become more interesting.