Let $(H, (\cdot,\cdot))$ be infinit dimensional separable Hilbert space. Also considerer $T : H \rightarrow H$ a non-null compact, self-adjoint operator such that $$ (T(v),v) \geq 0, \forall v \in H.\hspace{2cm} (1) $$ Denoting by $EV(T)$ the set of eigenvalues of $T$, we have $$ EV(T) = \{\lambda_n\}_{n=1}^{\infty}, \text{ with } \lambda_n \rightarrow 0 \text{ and } \lambda_n > 0, \forall n. $$ The fact that $\lambda_n > 0$ cames directlly from (1). The others facts relies on the fallowing theorems from H. Brezis, Functional Analysis:
Page 164 - Theorem 6.8,
Page 167 - Corollary 6.10,
Page 167 - Theorem 6.11,
Page 160 - Theorem 6.6 (Fredholm alternative) item a).
I would like to know if we can make sure the sequence of eigenvalues $\{\lambda_n\}$ is nonincreasing (all the sequence).
It's clear there's a subsequence with this properties. But, in some problems related to laplacian operator, they always use spectral theory to garantee the existence of such sequence and always seems they use results like the one a wrote.