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


*

*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?

*If the answer of the above question is assertive, does it hold even if - on a general C* algebra, or any von Neumann algebra?

*Can we define a Ky fan norm as we do for matrices?


For matrices of course the thing holds.   
 A: 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 $A$". 
