Is there any conclusions generalized Singular Value Decomposition into Hilbert Space  Spectrum decomposition can be regarded as the generalizations of the following fact that:
Every Hermitian  matrix $A$ can be decomposed into $A=U^{*}\Lambda U$,where $U$ is a unitary matrix
Singular vector  decomposition  can be expressed as Every Matrix $A_{mn}$ can be decomposed in to $A=U\Lambda V^{*}$, where $U$,and $V$ are unitary matrices.
Does it can be extended in to decompostion of linear operators on Hilbert Space. ?
I searched the internet and several traditional books about the topic "Singular Vector Decomposition into Hilbert space", However, to my disappointment, I find no similar conclusion.  Thanks for your help.
 A: This should really be a comment, not an answer, but I decided it perhaps could do with extra visibility.
The current version of your question, asks, among other things, once we have corrected the terminology:

Do operators on Hilbert space have a decomposition $A=U\Lambda V^*$ where $U$ and $V$ are unitary and $\Lambda$ is diagonal?

Leaving aside the subtlety about different versions of "diagonal" for operators in infinite dimensions, let me just note that if an injective operator $A$ is of the form $U\Lambda V^*$ where $\Lambda$ is diagonal and $U$ and $V$ are invertible, then it must have dense range; this is an easy exercise. The forward shift on $\ell^2({\mathbb N})$ is a simple example of an injective operator on Hilbert space that does not have dense range.
So the answer to your question, at least in the general form you have posed, is "no".
A: The simplest generalization is that a "compact self adjoint linear operator" on a Hilbert space can be diagonalized in terms of (an infinite number of) eigenvalues and eigenfunctions that are elements of the Hilbert space.  This can be extended to non-compact but still self adjoint operators, but it's more complicated because you might not have a discrete spectrum.  It can also be extended to a kind of singular value decomposition for compact but non self adjoint operators.   All of this is covered in textbooks on (linear) functional analysis.   
A: I believe what you are looking for is here:
http://mathworld.wolfram.com/PolarDecomposition.html
