The condensed problem:
I have a bounded, compact, self-adjoint, linear operator $A$ on $L^2([a,b];\mathbb{R})$ with positive eigenvalues $\{\frac1{\lambda_i}\}_i$.
Let $\lambda > 0$. Is there an operator $\tilde{A}$ with eigenvalues $\{\frac1{\lambda_i + \lambda}\}_i$? Does it have the same eigenfunctions? Is it bounded, compact, self-adjoint and linear?
My actual problem (maybe the details are helpful):
I have $A \phi = \int_a^b G(|t-s|) \phi(s) ds$ on $L^2([a,b];\mathbb{R})$, where $G\colon[0,T]\to \mathbb{R}$ is non-vanishing, continuous and positive definite, i.e. $$ \int_a^b \int_a^b G(|t-s|) \phi(t) \phi(s) ds dt > 0 $$ for all $\phi \neq 0$. By a small extension of Mercer's theorem, all eigenvalues $\{\frac1{\lambda_i}\}_i$ are positive, and the corresponding eigenfunctions $\{\phi_i\}_i$ are continuous.
I have expressions of the form $$ \sum_i \frac1{\lambda_i + \lambda} \langle \phi, \phi_i\rangle \phi_i $$ and would like to conclude with the Hilbert-Schmidt theorem that these are equal to $\tilde{A} \phi$ for some nice operator $\tilde{A}$.