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.



Is there any conclusions generalized Singular Value Decomposition into Hilbert Space 
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. 


