Resolvent

The characteristic polynomial $P_K(\lambda)$ of a finite-rank operator $K$ is given by:

$$ P_K(\lambda) = \det(K - \lambda I) $$

The roots of this polynomial are the eigenvalues of the operator $K$. These eigenvalues form the spectrum of $K$. Using these eigenvalues, the operator $K$ can be decomposed as:

$$ K = \sum_{i=1}^n \lambda_i P_i $$

where $P_i$ are the projections onto the eigenspaces corresponding to $\lambda_i$.

Resolvent and Operator Theory

The resolvent $R(\lambda, K) = (K - \lambda I)^{-1}$ solves the equation $(K - \lambda I)x = y$ for $y$ in the Hilbert space and $\lambda$ not an eigenvalue of $K$.

Eigenvalues

The eigenvalues of $K$ are identified by the roots of the characteristic polynomial $P_K(\lambda) = \det(K - \lambda I)$.

Functional Calculus

The resolvent defines functions of the operator $K$. For instance, if $f$ is a function, $f(K)$ is defined in terms of the resolvent:

$$ f(K) = \frac{1}{2\pi i} \int_\Gamma f(\lambda) R(\lambda, K) d\lambda $$

where $\Gamma$ is a contour enclosing the spectrum of $K$. This defines operations such as exponentiation, logarithms, and more for $K$.