Spectral-index source balls for compact operators

Assume Countable Choice. Let $X$ and $Y$ be real or complex Hilbert spaces over the same scalar field and let $K:X\to Y$ be compact. Write $\kappa :=\mathrm{\parallel }K\mathrm{\parallel }$, $S:=K^{\ast }K$, and $\mathrm{\mid }K\mathrm{\mid }:=S^{1\mathrm{/}2}$, and let ${\mathrm{\Lambda }}_K$ be zero together with the positive eigenvalues of $\mathrm{\mid }K\mathrm{\mid }$. An admissible spectral index is a bounded function $\psi :{\mathrm{\Lambda }}_K\to [0,\mathrm{\infty })$ with $\psi (0)=0$. Define $\psi (\mathrm{\mid }K\mathrm{\mid })$ by the compact spectral multiplier of $S$ with scalar function $t\mapsto \psi (\sqrt{t})$. For $\rho >0$ define $X_{\psi ,\rho }(K):=\{\psi (\mathrm{\mid }K\mathrm{\mid })w:w\in X,\mathrm{\parallel }w{\mathrm{\parallel }}_X\le \rho \}$. For $\nu >0$, ${\psi }_{\nu }(t)=t^{\nu }$ defines the power source ball $X_{\nu ,\rho }(K)$. If $\kappa >0$ and $q>0$, define the logarithmic spectral-index candidate by ${\psi }_{\log ,q}(0)=0$ and ${\psi }_{\log ,q}(t)=(1+\log (\kappa \mathrm{/}t){)}^{-q}$ for positive $t\in {\mathrm{\Lambda }}_K$; when it is bounded, it defines $X_{\log ,q,\rho }(K)$.

Axiom of Countable Choice

Absolute value of a compact operator

Compact spectral multiplier and source-power convention

Compact Hilbert-space inverse model

Operator norm

Clason, Regularization of Inverse Problems, Section 4.2

Hofmann and Mathé, A Note on the Modulus of Continuity, Section 1