Husemoller: homotopy of linear clutching map (proposition $4.5$, pag. $187$)

Question

Background :

I'm currently studying vector bundle through the book of [husemoller,"fibre bundles"] (https://www.maths.ed.ac.uk/~v1ranick/papers/husemoller). The following question concerns a detail about a quite specific proof, which I'm going to simplify for the sake of comprehension. However, the pages I'm reffering to are $147-148$, proposition $4.6$.

Let's define a linear clutching map as map of the form $p(x,z) = a(x)z+b(x)$, invertible for $|z| \in S^1$. This a continuos family of automorphism of the fibre over $x$ of a vector bundle $\zeta$ over $X$ (compact connected $CW$, if needed).

Let's also take in account $p_0(x)$ and $p_{\infty}(x)$ two linear maps from $\zeta_x \to \zeta_x$ such that $p_0^2 = p_0$ and $p_{\infty}^2 = p_{\infty}$(which have an explicit form that I'd like to avoid for the moment if possible). Let's recall that with the "involution" property, it holds that the vector space, in this case we may assume $\mathbb{C}^n$ is the direct sum of kernel and image in both cases.

Proposition: $p(x,z)p_0(x) = p_{\infty}(x)p(x,z)$.

The proposition give rise to two maps $p_{+} : im p_0 \to im p_{\infty}$ and $p_{-} : ker p_0 \to kerp_{\infty}$, which are the restrictions of $p(x,z)$.

There is also the followinf proposition:

Proposition $(4.5)$: $p_{+} : im p_0 \to im p_{\infty}$ is an isomorphism for $|z| \geq 1$ while $p_{-} : ker p_0 \to kerp_{\infty}$ is an isomorphism for $|z| \leq 1$.

The background ends here.

Problem :

Now the problem is the following proposition:

Proposition: Let $p_{+},p_{-}$ be defined as above. and let $p^t := > p_{+}^t+p_{-}^t$ where $p_{+}^t := a_{+}z+tb_{+}$ and $p_{-}^t:= ta_{-}z+b_{-}$ for $0 \leq t \leq 1$. Then there is a homotopy of linear clutching maps from $a_{+}z+b_{-}$. Moreover, the bundles $[\zeta,p] \simeq [im p_0,z]\oplus [ker p_0,1]$

The proof given is the following:
$\hspace{3.5cm}$

Questions :

1. Why $p_{+}^t,p_{-}^t$ are isomorphism on their images for $0 \leq t\leq 1$? I can see an argument of rescaling by $t$ for $p_{-}$ since $|zt| \leq 1$ for $|z| \leq 1$ although I see a problem diving by $t$ in $p_{+}^t$ this rescaling would work for each $t \ne 0$ but in $t=0$ I think I have a problem, or there is a way to prove that $a_{+}$ is invertible apriori? Note that I'm taking $\{\infty\} \in \{|z| \geq 1\}$ so that $\{\infty\} \in \{|z| \geq 1\} \cup \{|z| \leq 1\}$, perhaps it could help.

2. Why should be true that $a_{+}$ and $b_{-}$ are isomorphism?

3. Why the linear combination i.e the homotopy, which can be re-written as $(1-t)(a_{+}(x)z+b_{-}(x))+tp(x,z)$ is a linear clutching map (i.e invertible) for each $0 \leq t \leq 1?$ For $t = 0$ I don't see how $a_{+}(x)z+b_{-}(x)$ is even defined, since $a_{+}$ and $b_{-}$ are part of the restriction to $a_{+}$ and $b_{-}$ respectively, should take vector in the intersection of the domain, but $ker p_0 \cap im p_0 = \{0\} $ since they are in direct sum.

Any help or reference in order to clarify this details would be appreciated, thanks in advance.