I am wondering if I have a sequence $\{ K_i \}_{i=1} ^\infty$ such that each $K_i: H\rightarrow H $ where $H$ is a separable Hilbert space. Then what are some criteria/ definitions for linear independence of the operators?
How I understood linear independence of operators was that $A$ and $B$ are linearly independent if $\langle Ax,x \rangle \not = \alpha \langle Bx,x \rangle$. But that was my understanding only from a asking google, and it seems to be only pairwise one a good criteria for a sequence.
According to your definition, an operator and its dual would always be linearly dependent, since $$\forall x, \left<Ax,x\right>=\left<x,A^*x\right>$$
The correct definition is that $A$ and $B$ are linearly independent if there is no couple $(\alpha,\beta)\neq (0,0)$ such that $$\forall x, \alpha A(x)+\beta B(x)=0$$
For a larger family, this definition extends straightforwardly: $\{A_i\}_{i=1}^n$ is linearly independent iff there is no collection $(\alpha_i)_{i=1}^n$ not all zero such that $$\forall x, \Sigma \alpha_i A_i(x)=0$$