Let $L^2$ be a separable Hilbert space and let $\{e_i(t)\}_{i \in \mathbb{N}}$ be a basis for it. Moreover let $\phi(T,z)$ and $\psi(T,z)$ be 1-parameter families (in z) in $L^2$, with basis expansions $$ \phi(T,z) = \sum_i c_i(z)e_i(T), \, \psi(T,z) = \sum_i \tilde{c}_i(z)e_i(T) $$
Then is the operator define by $$ A(z)^{+}c_i(z)=\tilde{c}_i(z) $$ continuous in z, where $c_i^+$ denotes the Moore-Penrose pseudo inverse. It seems so to me but I am not certain about how to prove this?
