Characterizations of diagonalizability of operators on infinite-dimensional vector spaces I would like to understand the subset $\mathcal{B}_{\mathrm{diag}}(\mathcal{H}) \subset \mathcal{B}(\mathcal{H})$ of diagonalizable operators on some potentially infinite-dimensional Hilbert space better. Specifically, I would like to prove the technical lemma below.
Sometimes, diagonalizable means “unitarily diagonalizable”, i. e. normal with respect to the adjoint ${}^*$ of the given scalar product with $\langle \, \cdot \, , \, \cdot \, \rangle$. But I have the more general meaning in mind where you replace unitary with invertible:
Equivalent Characterizations of Diagonalizability

*

*$H \in \mathcal{B}(\mathcal{H})$ is diagonalizable.


*$H$ admits a functional calculus for bounded Borel functions $f : \mathbb{C} \longrightarrow \mathbb{C}$.


*There exists a positive, bounded, invertible operator $W = W^* \in \mathcal{B}(\mathcal{H})$ with bounded inverse $W^{-1} \in \mathcal{B}(\mathcal{H})$ so that $H$ is normal with respect to the weighted scalar product $\langle \varphi , \psi \rangle_W := \langle \varphi , W \psi \rangle$.


*$H = H_{\Re} + \mathrm{i} H_{\Im}$ is the sum of two bounded commuting operators that are selfadjoint with respect to some weighted scalar product $\langle \varphi , \psi \rangle_W := \langle \varphi , W \psi \rangle$.
I could add other characterizations (e. g. that $H$ is related to a multiplication operator by a similarity transform), but for my purposes, I think this list suffices. First of all, is this list of characterizations correct?
At the end of the day I would like to prove the following:
Lemma
If $\lambda \mapsto H(\lambda) \in \mathcal{B}_{\mathrm{diag}}(\mathcal{H})$ is a continuous path in the set of diagonalizable operators, then there exists a continuous positive-selfadjoint-operator-valued map $\lambda \mapsto W(\lambda) \in \mathcal{B}(\mathcal{H}) \cap \mathcal{B}(\mathcal{H})^{-1}$ so that $H(\lambda)$ is normal with respect to $\langle \, \cdot \, , \, \cdot \, \rangle_{W(\lambda)}$.
Everything except for the continuity of $\lambda \mapsto W(\lambda)$ seems clear to me. That is because the choice of $W$ is not unique — for matrices, for example, you are free to choose units of length for each of the eigenvectors and $W$ declares them to all have length $1$.
 A: For future reference, I came across volume 3 of Dunford and Schwartz's book series on linear operators, which is dedicated to spectral operators. These are a generalization of what I called diagonalizable operators; more precisely, they are in the language of Dunford & Schwartz scalar operators on a Hilbert space (cf. Chapter XV.6.4, Theorem 4).
A spectral operator $H$ is an operator on a Banach space with a projection-valued measure $\{ P(\Lambda) \}_{\Lambda \in \mathfrak{B}(\mathbb{C})}$ over the Borel sets $\mathfrak{B}(\mathbb{C})$ of the complex plane (oblique-projection-valued as there is no Hilbert space adjoint), which (a) commute with the operator (and therefore, the spectral subspaces are left invariant) and (b) where the spectrum of the restricted operator $\sigma \bigl ( H \vert_{\mathrm{ran} \, P(\Lambda)} \bigr ) \subseteq \overline{\Lambda}$ (Chapter XV.2.2, Definition 2).
The projection-valued measure of a spectral operator is unique (Chapter XV.3.5, Corollary 8).
This leads to a unique decomposition (Chapter XV.4.3, Theorem 5) of a spectral operator
\begin{align*}
H &= S + N
\end{align*}
into a scalar part (Chapter XV.4.1, Definition 1),
\begin{align*}
S &= \int_{\mathbb{C}} \lambda \, \mathrm{d} P(\lambda) 
\end{align*}
and a quasi-nilpotent or residual part,
\begin{align*}
N &= H - S 
. 
\end{align*}
I have just read the beginning, but Dunford & Schwartz seem to develop the basic theory of spectral operators in the book over the course of almost 700 (!) pages. Their work was totally under my radar, so I hope if someone looks for “diagonalizable” in the future, they might find their book useful. Dunford also has a shorter article on that subject, which summarizes the basic definitions and properties.
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Reference: Linear Operators, Part III, Spectral Operators
Nelson Dunford, Jacob T. Schwartz, William G. Bade and Robert G. Bartle
Pure and Applied Mathematics, Volume VII
Wiley-Interscience, 1971
