I am currently writing an article relating to multiplicative ergodic theorems for cocycles of bounded operators acting on a Hilbert space, and in parts of the argument it is necessary for me to refer to exterior powers of Hilbert spaces and to singular values of bounded linear operators on Hilbert spaces. Since the target audience of the paper may not be very familiar with these topics I am writing a few short expository paragraphs in order to familiarise the audience with the necessary material, and I am struggling a little to find references for some of the topics in question.
So far I have found the fifth chapter of Roger Temam's book Infinite-dimensional Dynamical Systems in Mathematics and Physics a useful source of information on the construction of the exterior powers of a Hilbert space, and to some extent for singular values of operators on a Hilbert space as well, which it defines via the formula $$s_k(L):=\sup_{\substack{F\subset H\\\dim F=k}} \inf_{\substack{v \in F\\\|v\|=1}} \|Lv\|$$ for the $k^{\mathrm{th}}$ singular value of the bounded linear operator $L$. Here the supremum is taken over all $k$-dimensional subspaces $F$ of the Hilbert space $H$. However I lack a reference for the following important fact:
- Let $L$ be a bounded operator on the Hilbert space $H$. Then the second singular value of the bounded operator $\wedge^kL$ on the Hilbert space $\wedge^kH$ is given by the formula $$s_2(\wedge^kL)=\left(\prod_{i=1}^{k-1}s_i(L)\right)s_k(L).$$
Is anyone able to suggest a crisp reference for this particular fact? I have not even found a decent reference for this in the case where $H$ is finite-dimensional, although in that context it is much easier to prove it from first principles since the singular values correspond to eigenvalues of $L^*L$.
I also have some lower-priority items on my wish list:
I would ideally like to have crisp references for the following identities. Let $L_1,L_2$ be bounded operators on the Hilbert space $H$. Then $s_k(L_1L_2) \leq \|L_1\|s_k(L_2)$, $s_k(L^*_1)=s_k(L_1)$ and $s_k(L_1L_2)\leq s_k(L_1)\|L_2\|$, for every $k \geq 1$. (The identity $s_k(L_1L_2) \leq \|L_1\|s_k(L_2)$ in fact follows directly from the definition of the singular values, but I have not found an equally simple proof of the other two: using the identity $$\|\wedge^kL\|=\prod_{i=1}^{k}s_i(L)$$ which I can find in Temam's book, it is is possible to deduce the identity $s_k(L^*_1)=s_k(L_1)$ by combining this identity with the fact that $\|L^*\|=\|L\|$ for every bounded operator $L$ on $\wedge^kH$. The third fact $s_k(L_1L_2)\leq s_k(L_1)\|L_2\|$ can then be obtained by combining the first two. This is I suppose a viable way for me to present these identities to the reader but a crisp reference would be nicer.)
I would also be delighted if anyone can suggest other references than Temam's book for the construction of the exterior powers $\wedge^kH$.
I am aiming to achieve the maximum possible generality in my article, so I would like to allow $H$ to be either real or complex, finite- or infinite-dimensional, separable or non-separable. Similarly I wish to make no assumptions whatsoever on the operators $L$ beyond the fact that they are bounded linear operators acting on $H$.
Thank you all in advance!
