Equivalent characterization of compactness for a bounded operator This question is related to a question I asked yesterday: Toeplitz operator on Bergman space
Consider a linear bounded operator $T: H \rightarrow H$, where $H$ is a Hilbert space of holomorphic functions on the unit disc $\mathbb{D}$ in $\mathbb{C}$ (in my case, even a reproducing kernel Hilbert space), and suppose that the unit ball in $H$ is a normal family (every sequence of the unit ball in $H$ has a subsequence that converges uniformly on compact subsets of $\mathbb{D}$). Is it true that 

$T$ is compact on $H$ if and only if every bounded sequence $(f_n)$ in $H$ with $$f_n \rightarrow 0$$ as $n \rightarrow \infty$ uniformly on compact subsets of $\mathbb{D}$ has a subsequence $(f_{n_k})$ such that $$Tf_{n_k} \rightarrow 0$$ as $k \rightarrow \infty$ in $H$?

I am also interested in references where I may find similar results. 
 A: This characterization of compact operator can be derived by a more general one. Let us take in general $T:E\to F$ a linear bounded operator between two Banach spaces. We will say that $T$ is compact if the image $T(B_E)$ of the closed ball of $E$ is relatively compact in $F$. We will say that $T$ is sequentially compact if: $x_n \rightharpoonup x$ weakly in $E$ implies that a subsequence $T(x_{n_k})\to T(x)$ in F. All this theory can be found in Brezis - Functional Analysis (maybe under the form of exercise).
If $T$ is compact, then it is sequentially compact. In fact take $x_n \rightharpoonup x$ weakly in $E$, then $x_n$ is bounded, that is it is contained in a ball in $E$. Then the image of the ball is relatively compact in $F$, then we get a subsequence $T(x_{n_k})\to y$ in $F$. And by continuity with respect to weak convergence we have $y=T(x)$.
If $E$ is reflexive (that is the case of a Hilbert space), if $T$ is sequentially compact, then it is compact. In fact: since $E$ is reflexive, the unit ball $B_E$ is weakly compact, then for each $\epsilon_n>0$ there is a finite cover of the ball made of weak neighborhoods:
$$V_{\epsilon_n}(x_i^n;f_1,...,f_{r_i}^n):=\{x\in B_E:|f_l(x-x_i^n)|<\epsilon_n\,\,\forall l=1,...,r_i  \}\qquad i\in I^n,$$
where each $f_l\in E^*$, $r_i\in\mathbb{N}$ and $I^n$ is finite. Taking a sequence $\epsilon_n\searrow0$ we get that the set $\{T(x_i^n) \}_{n\in\mathbb{N},i\in I^n}$ is dense in the closure of $T(B_E)$; in fact if $y\in T(B_E)$, then $y=T(x)$ and there is $x_j\in\{x_i^n\}$ such that $x_j \rightharpoonup x$, then by hypothesis of sequential compactness $T(x_j)\to y$ unless passing to subsequence. Thus we have proved that the closure of $T(B_E)$ is separable in $F$, since $\{T(x_i^n)\}_{n\in\mathbb{N},i\in I^n}$ is countable. Now we show that $T(B_E)$ is sequentially relatively compact in $F$ (that together with separability gives relatively compactness in the strong topology of $F$). In fact let $T(x_n)$ be a sequence in $T(B_E)$, then $x_n\in B_E$, then a subsequence $x_{n_k} \rightharpoonup x$ by weak compactness, then by hypothesis a subsequence $T(x_{n_{k_l}})\to T(x)$ in $F$.
Now we come to the question. I reasonably assume that uniform convergence on compact sets implies weak convergence in $H$ (this is usually true in the usual Hilbert spaces). And we take $T:H\to H$.
If $T$ is compact, take $f_n$ bounded in $H$ and uniformly converging to zero on compact sets. Then it weakly converges to zero in $H$. By sequential weak compactness of the operator we get $T(f_{n_k})\to 0$ in $H$.
Now assume $T$ satisfies the right hand of the claimed "if and only if". We show that $T$ is sequential compact, that will imply compact. Take a sequence $f_n\in H$ that is weakly converging to $f$. Then $g_n:=f_n-f$ is weakly converging to $0$. Hence it also bounded, then by the hypothesis of "normal family" in the question, $g_n$ uniformly converges on compact sets (after passing to subsequence). By uniqueness the uniform limit of $g_n$ is zero, since this is the weak limit. By assumption a subsequece have $T(g_{n_k})\to0$ in H, and by definition of $g_n$ we have $T(f_{n_k})\to T(f)$ in $H$, that proves sequential compactness of the operator, and then its compactness.
