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I have a question on (orthogonal) direct sums of an Operator.

In particular, I was wondering if the direct summands of a linear and bounded operator $T$ on a complex Hilbertspace $H$ are all $T$-reducible closed subspaces, where a subspace $M$ is reducible if $T(M)\subset M$ and $T(M^\perp)\subset M^\perp$. Is that true, and if so, why?

I don't know much about direct sums of operators, and the statement I proposed is what I was able to derive from here: Formal definition of direct sum of operators.

thanks in advance

BabyWienerSpace
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  • It is only a matter of choice in the definition of what you call "direct summands of a linear and bounded operator on a complex Hilbert space". – Anne Bauval Jan 21 '23 at 13:22
  • How would you define it? I read it in a paper and was quite confused about it. If I choose to say that the "direct summands of $T$" are precisely the $T$-reducible subspaces everything that follows in the paper worked out quite fine. So I kind of searched a suitable definition of "direct summand" which is equivalent to what I need, namely being $T$-reducible. – BabyWienerSpace Jan 21 '23 at 13:33
  • I believe you were right to choose that definition, i.e. in the context of Hilbert spaces, I think that "direct sum" implicitely means "orthogonal direct sum". – Anne Bauval Jan 21 '23 at 13:42

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