Let $H$ be a Hilbert space and $A:D(A)\subset H \rightarrow H$ a linear operator. In what sense are the deficiency indices
$$\mathrm{dim}(\mathrm{ran}(A+ z) )\text{ }(z\in \mathbb{C})$$
defined?
I mean how to make sense of the dimension here?
Is it the number of basis vectors where the basis is defined as a Schauder basis of orthonormal vectors?
My motivation to aks this:
Two Hilbert spaces are isomorphic iff they have the same Hilbert dimension. Hilbert dimension is defined in terms of a Schauder basis of orthonormal vectors. Using this fact one can show that a symmetric and closed operator has a selfadjoint extention iff
$$\mathrm{dim}(\mathrm{ran}(A+i)^\perp)=\mathrm{dim}(\mathrm{ran}(A-i)^\perp)$$
I doubt that this would hold when we use a different notion for the dimension.