Extension beyond compact resolvent via regularization

Let $X_1,\ldots,X_r$ be spectral operators in the sense of Dunford on separable Hilbert spaces $\mathcal{H}_1,\ldots,\mathcal{H}_r$. Assume there are positive self-adjoint operators $K_1,\ldots,K_r$ with compact resolvent such that $X_{j,\varepsilon}=X_j+\varepsilon K_j$ satisfy the hypotheses of the conditional regularization-convergence theorem for all sufficiently small $\varepsilon>0$. Let $\widetilde{X}_1,\ldots,\widetilde{X}_r$ be their tensor liftings on

$$\mathcal{H}_{\otimes}=\mathcal{H}_1\otimes\cdots\otimes\mathcal{H}_r.$$

Define the lifted spectral measures and nilpotent components by

$$d\widetilde{E}_j(\lambda_j)=I_1\otimes\cdots\otimes dE_{X_j}(\lambda_j)\otimes\cdots\otimes I_r,$$

and

$$\widetilde{N}_j(\lambda_j)=I_1\otimes\cdots\otimes N_j(\lambda_j)\otimes\cdots\otimes I_r,$$

and analogously for the regularized operators $X_{j,\varepsilon}$. Extension beyond compact resolvent via regularization. The contour-integral representation of the tensor-lifted functional calculus should be well-defined for every holomorphic function $f$ on an open polydisk containing the product spectrum, and

$$\operatorname*{s-\lim}_{\varepsilon\to0}f_{\otimes}(X_{1,\varepsilon},\ldots,X_{r,\varepsilon})=f_{\otimes}(X_1,\ldots,X_r)$$

in the strong operator topology. If the resolvent convergence is norm resolvent, the limit should hold in operator norm with explicit error bounds. The explicit projector--nilpotent expansion, including the terms with $A\ne\emptyset$, should also remain valid for the limit operators, subject to the additional spectral stability assumptions required for convergence of the projectors and nilpotent components. This conjectural extension would broaden the unified tensor-lifted functional calculus from the compact-resolvent setting to general unbounded non-self-adjoint spectral operators. The conditional convergence theorem establishes the contour-integral convergence once its regularization hypotheses hold, but the required hypotheses and the stability of the pointwise projector--nilpotent structure, especially for continuous spectrum, remain open.