Resolvent of an element ($R(\lambda ,a)$)

The resolvent is $R(\lambda ,a)=(\lambda 1-a{)}^{-1}$ on the complement of the spectrum of an element. In a Banach algebra it is analytic there.

For a complex unital Banach algebra and a circle enclosing its element's Banach algebra spectrum, the displayed identity holds for $n\ge 0$. On a sufficiently large circle, expand the Neumann series and integrate it term by term. At any resolvent point the local series $R({\lambda }_0+h)={\sum }_{j\ge 0}(-h{)}^{j}R({\lambda }_0{)}^{j+1}$ proves norm analyticity. Apply continuous complex linear functionals and the scalar Cauchy theorem to deform the large circle to any radius greater than the maximum spectral modulus; the Hahn-Banach theorem separates points and restores the vector integral identity. Its norm bound $\Vert x^n\Vert \le R^{n+1}{\max }_{|\lambda |=R}\Vert R(\lambda )\Vert$ gives the reverse inequality in the spectral radius formula.

For two points $z,w$ in the resolvent set of an operator $A$,In particular, the resolvent operator is differentiable and ${\partial }_{z}R(z,A)=-R(z,A{)}^2$.

For a closed operator, the resolvent is operator-valued holomorphic on its resolvent set. On a Hilbert space its norm is subharmonic and therefore cannot have a strict interior maximum.

The $ϵ$-pseudospectrum consists of spectral points together with resolvent points where $\Vert (A-zI{)}^{-1}\Vert >{ϵ}^{-1}$. Equivalently it is described by a small lower norm.

For a spectral subset isolated by a contour $\Gamma$ in the resolvent set,projects onto its spectral subspace. Norm-small perturbations preserve the rank of this projection.

For a bounded operator on a complex Banach space, partition a disconnected spectrum into two nonempty compact pieces. The holomorphic function equal to one near the first piece and zero near the second defines a Riesz projection. The holomorphic spectral mapping theorem makes it neither zero nor identity. Its closed range is a proper nonzero invariant vector subspace. This need not be an orthogonal projection, and the argument does not require a normal operator.