Holomorphic operators, what does it mean

Question

It is clear for me what means that a function is holomorphic, but what about operators? What does it mean that an operator is holomorphic?

In particular, consider the free Dirac operator
$$ H_0 = -i\alpha\cdot\nabla + m\beta,$$
with $m\in\mathbb{R}$ and $\alpha, \beta$ are just Hermitian $4\times 4$ matrices.

Is this operator holomorphic? Could anyone explain me why or give me some references?

Thank You in advance!

Answer 1

The expression is presumably taken from Kato's Remarks on Holomorphic Families of Schrödinger and Dirac Operators, where he talks not about holomorphic operators but about "holomorphic families $T(κ) = T + κA$ of linear operators in a Hilbert space". For definitions we are sent to Kato's book Perturbation theory for linear operators, ch. VII.1.2. When the operators in the family are bounded "holomorphic" means just what one would expect, i.e. $T(κ)$ is complex differentiable (strongly or weakly, it does not make a difference which). Kato calls this "bounded-holomorphic". But since the Schrödinger and Dirac operators are unbounded the definition becomes more complicated and technical:

"A family of operators $T(κ)\in\mathcal{C}(X,Y)$ defined in a neighborhood of $κ = 0$ is said to be holomorphic at $κ = 0$ (in the generalized sense) if there is a third Banach space $Z$ and two families of operators $U(κ)\in\mathcal{C}(Z,X)$ and $V(κ)\in\mathcal{C}(Z,Y)$ which are bounded-holomorphic at $κ = 0$ such that $U(κ)$ maps $Z$ onto $D(T(κ))$ one to one and $T(κ)U(κ)=V(κ)$. $T(κ)$ is holomorphic in a domain D of the complex plane if it is holomorphic at every $κ$ of $D$."

This is motivated by the notion of generalized convergence of closed operators introduced in chapter IV of the same book which generalizes to them the notion of convergence by norm of bounded operators. As for type (A), we find that in VII.2.1:

"A family $T(κ)\in\mathcal{C}(X,Y)$, defined for $κ$ in a domain $D_0$ of the complex plane, is said to be holomorphic of type (A) if i) $D(T(x)) = D$ is independent of $κ$ and ii) $T(κ)u$ is holomorphic for $κ\in D_0$ for every $u\in D$."

Holomorphic type (A) families of Dirac operators are briefly discussed in section VII.3.3 of the book.