# Continuity concerning family of projections

Let $$X$$ be a compact topological space, $$H$$ be a complex Hilbert space and endow $$F(H)$$, the space of bounded Fredholm operators in $$H$$, with the uniform norm topology (inherited from $$B(H)$$).

Let $$T: X\to F(H)$$, $$x\mapsto T_x$$, be a continuous map. There exists a closed subspace $$V\subseteq H$$ of finite codimension, i.e. $$\dim H/V<\infty$$, such that $$V\cap \ker T_x = \{0\}$$ for all $$x\in X$$.

I have proved that $$H/T(V) = \bigsqcup\limits_{x\in X} H/T_x(V)$$ is a vector bundle over $$X$$ (of finite rank). In particular, $$\dim H/T_x(V)$$ is independent of $$x$$ (here we can assume connectedness of $$x$$).

For $$x\in X$$, let $$P_x: H\to H$$ be the orthogonal projection onto $$T_x(V)$$. In order to induce a specific map of bundles (see here for details), I need to check the continuity of the map $$X\times H\to H$$ given by $$(x,u)\mapsto P_x(u)$$.

Question: Is $$(x,u)\mapsto P_x(u)$$ continuous?

Looking at the inequality $$\|P_y(v)-P_x(u)\| \leq \|P_y(v-u)\| + \|(P_y-P_x)(u)\|$$ we conclude it suffices to prove that $$x\mapsto P_x$$ is continuous when one gives $$B(H)$$ the strong operator topology, but I could not prove it.

Any help is appreciated. Thanks in advance!

## 1 Answer

Let $$S_x=T_x|_V$$. Then $$S_x$$ coincides with the composition of $$T_x$$ with the inclusion of $$V$$ into $$H$$, both Fredholm operators, so $$S_x$$ is also Fredholm.

Like every Fredholm operator, $$S_x$$ has closed range and it is clearly one-to-one. From this and the open mapping Theorem it easily follows that $$S_x^*S_x$$ is invertible.

Therefore $$R_x:=S_x(S_x^*S_x)^{-1/2}$$ is a well defined isometry having the same range as $$T_x$$ and we deduce that $$P_x=R_xR_x^*$$, from where the norm continuity of $$P_x$$ follows.

• Thank you Ruy! I was trying to prove that $R$ is an isometry without success. In order to an operator $A:V_1\to V_2$ to be an isometry, it is necessary that $A^*A=id_{V_1}$. We have $R^*R=(S^* S)^{-1}$, which need not to be $id_{V}$. Am I missing something? – Rodrigo Dias Aug 14 '20 at 18:52
• @RodrigoDias, I am sorry. The exponent should be $-1/2$. I have just edited my answer to fix this. It should be like in the polar decomposition. – Ruy Aug 14 '20 at 19:57
• Right, I got it! – Rodrigo Dias Aug 14 '20 at 20:12