On singular value decomposition

Question

Let us consider any arbitrary Hilbert space $H$ (need not be of finite dimension). Consider a compact operator $A$ compact operator in $\mathcal{B}(H)$.

1. Does it always have a singular value decomposition? If so, is there any quick way to see the singular values and the unitaries which do this trick?
2. If the answer of the above question is assertive, does it hold even if - on a general C* algebra, or any von Neumann algebra?
3. Can we define a Ky fan norm as we do for matrices?

For matrices of course the thing holds.

Answer 1

A compact operator on a Hilbert space always has its spectrum consist entirely of eigenvalues, with the possible exception of zero: i.e., the spectrum is either finite (and it will include $0$ in that case) or it consists of a sequence that converges to zero.

Consider first a positive compact operator $A$. One can show that there is an orthonormal basis of eigenvectors, and so $$\tag{*}A=\sum_{n=1}^\infty\lambda_nP_n,$$ where $P_1,P_2,\ldots$ are rank-one projections, $\lambda_1\geq\lambda_2\geq\cdots \geq0$ for all $n$, and $\lambda_n\to0$. This is the singular value decomposition of $A$.

For an arbitrary compact operator $A$, we have the polar decomposition $A=VT$, where $V$ is a partial isometry and $T=|A|=(A^*A)^{1/2}$, which is compact. By the above, $T$ has positive eigenvalues, and these will be the singular values of $A$.

As the singular values are well-defined, it is certainly possible to define the Ky-Fan norms. The first Ky-Fan norm agrees with the operator norm. The "infinite" Ky-Fan norm will not be well-defined for all compact operators; those where it works are called "trace-class".

For arbitrary operators, the above does not apply; there are positive operators in $B(H)$ with no eigenvalues at all (but nonempty spectrum, though). The equality $(*)$, known as the Spectral Theorem, has a generalization in terms of integrals: $$ A=\int_{\sigma(A)}\lambda\,dE_A(\lambda), $$ where $E_A$ is a projection-valued measure.

Now, one can define singular values without the Spectral Theorem. One needs projections and a trace, though, so it is commonly done in semifinite von Neumann algebras. One defines $$ \mu_A(t)=\inf\{\|AP\|:\ \tau(I-P)\leq t\}=\min\{s:\ \tau(E_{|A|}(s,\infty)\leq t\}. $$ Here $\mu_A(t)$ is the continuous analogue, the "$t^{\rm th}$ singular value of $A$".