Endomorphisms of stable vector bundles over a Riemann surface Hello everyone, 
it seems to be "well-known" that $H^0(X;End(V))$ only contains isomorphisms where $X$ is a Riemann surface and $V$ a stable (algebraic) vector bundle over $X$. The usual proof considers (roughly) the (coherent) image sheaf of a non-zero vector bundle morphism $\varphi:V\to V$ and one obtains a contradiction under the hypothesis that $im(\varphi)\neq V$ : $\mu(V)<\mu(im(\varphi))<\mu(V)$  (where $\mu(V)=deg(V)/rk(V)$ denotes the slope of $V$). 
Now my question is if this assertion is still true in the complex (differential) geometric context, i.e. when one defines holomorphicity (and hence stability) via del-bar-operators. 
The previous proof doesn't seem to work since the image of a (smooth) vector bundle homomorphism $\varphi:V\to V$  is in general NOT a vector (sub)bundle of $V$ (unless the rank of $\varphi$ is (locally) constant). So $\mu(im(\varphi))$ doesn't make sense since the image of $\varphi$ is in general not a subbundle of $V$. 
Or am I missing something in the holomorphic setup? 
Since I'm not an expert in algebraic geometry I have some difficulties in "translating" results concering stable (algebraic) vector bundles over a Riemann surface into the complex geometric (del-bar) approach to stable (holomorphic) vector bundles. 
Thanks in advance for any answers, comments and remarks!
Edit: I changed my terminology from "differential geometric (dg)" to "complex geometric" which seems more appropriate (thanks @David Roberts!) and added some comments. Hopefully my questions/confusion is more understandable now. 
 A: Consider the two following potential definitions of stability of a locally free coherent sheaf $E$.
(A)  Every subbundle (i.e. locally free subsheaf) has strictly smaller slope.
(B)  Every subsheaf has strictly smaller slope.
I claim that over a curve $X$ these two definitions are equivalent.  Indeed, suppose (A) holds, and let $F\subset E$ be a subsheaf.  Consider the sequence
$$0\to F\to E \to Q\to 0.$$
Here $Q$ may fail to be locally free; Put $G = Q/Q_{tors}$, which is then locally free since $X$ is a curve.  Define $K$ by the sequence
$$0\to K \to E \to G\to 0,$$
and observe that $K$ is locally free of the same rank as $F$.  Furthermore, $c_1(K) \geq c_1(F)$ since $c_1(Q_{tors})\geq 0$, so $\mu(F) \leq \mu(E)$.
Thus it is equivalent to only consider subbundles for stability, so long as we are working on a curve.  Together with GAGA, this shows the two approaches (and any possible combination of definitions) are essentially the same.
A: Although this is rather a comment (or at best a partial answer) than an (honest) answer, I don’t write it as such. But it’ll contain some thoughts about my question and so it seems appropriate to write it as an answer (at least to me…). 
@Jason Starr: Of course you’re completely right. Sorry, I was probably too sloppy by only writing “Riemann surface” than “compact (connected) Riemann surface”.  Maybe I hoped that it would be clear from the context.
In any case, your (and Jack’s) comment sparked my interest in GAGA and how it really applies. 
@YangMills: Today I was able to read some passages of Kobayashi’s “Differential geometry of complex vector bundles” and it clarified some of my confusions. Thanks!
So here are my thoughts about my question: 
Let $X$ be a compact Riemann surface and denote by $O_X$ the sheaf of holomorphic functions on $X$. Then there is a bijection between holomorphic vector bundles of rank $r$ over $X$ and locally free $O_X$-modules of rank $r$. But what are the morphisms on each side? 
The morphisms of locally free $O_X$-modules (of rank $r$) are simply sheaf homormorphisms.
Maybe all authors (at least for example Huybrechts or Kobayashi) define a holomorphic vector bundle morphism $\varphi:V\to W$ (where $V$ and $W$ are holomorphic vector bundles over $X$) as a holomorphic map $\varphi:V\to W$ such that the restriction of $\varphi|x$ to any fiber $V(x)$ maps complex linearly to $W(x)$ AND the rank of $\varphi|x$ is independent of $x\in X$ (call this definition “definition B”).
In my question I used an alternative definition (“definition A”) of a holomorphic vector bundle morphism, i.e. I didn’t require the rank to be constant. If one uses the definition B instead, the image of a holomorphic vector bundle morphism  $\varphi: V\to V$ is indeed a holomorphic subbundle of $V$ and one can proof that $H^0(End(V))=H^0(Iso(V))$ just as in the “sheaf case”. 
But with definition A one can prove this fact as well by using sheaf theory because we can define stability (resp. slope $\mu(P)$) for every torsion-free coherent sheaf $P$ over $X$. Without going into detail I shall just mention that stability for $P$ means that $$\mu(P')<\mu(P)$$ for every coherent subsheaf $P'$ with $0 < rank(P') < rank(P)$.  
Now every holomorphic vector bundle homomorphism $\varphi:V\to V$ (w.r.t definition A) induces a sheaf homorphism $\phi:S\to S$ of the underlying locally free $O_X$-sheaf $S$ and vice versa. (This fails in definition B.) The image of $S$ under $\phi$ is in general no longer locally free (claim: this is precisely the case when $\varphi$ has constant rank). But we can construct the (coherent) image sheaf $im S$ of $S$ so that the slope of $im S$ is defined. Then we can proceed as I indicated in my question (cf. Kobayashi’s “Differential geometry of complex vector bundles”) and we see that $\phi$ and hence $\varphi$ is an isomorphism.
Even if this seems quite plausible I’m still not really satisfied because the above stability condition seems to be much stronger than the one for a holomorphic vector bundle $V$ since one considers not only locally free subsheaves (i.e. holomorphic subbundles of $V$) but also coherent subsheaves of the corresponding locally free sheaf.  
So my feeling is that if one uses definition A (i.e. omitting constant rank) then $H^0(End(V))=H^0(Iso(V))$ is in general only true under this stronger stability condition. 
(But it’s also possible that this is complete non-sense…) 
