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20:01,30 2010-10-25. You’ll receive a confirmation message: A confirmation message is sent back to your e-mail address just. Look at the bottom of the SPSC organization home to this year, we have never had so.Q:
Spectral decomposition of a matrix of operators, can we use orthonormality
I have a question that I am struggling to answer.
Let $T_1$ and $T_2$ be two bounded linear operators, acting on a Hilbert space $\mathcal{H}$.
Define: $A=T_2T_1-T_1T_2$. Can we then say that $$A=\sum_i\lambda_iP_i,$$ where each $P_i$ is an orthonormal projection, and $\lambda_i$ are eigenvalues of $A$?
I have tried to apply the Spectral theorem, but the eigenvalues do not seem to match up. For instance, take $T_1=T_2=Id$, and $T_1T_2-T_2T_1=0$. This clearly implies $0$ is an eigenvalue of $A$. The eigenvalue is clearly $\lambda=0$ but if I try and expand this to an orthonormal basis, it should be: $0.\phi_i,\psi_i$.
A particular case, the uniqueness of the spectral theorem, is discussed here:
Uniqueness of spectral decomposition for closed self adjoint operator
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