Consider an operator which is defined in Banach or Hilbert space by a 2x2 matrix with entries A, B, C, D which where linear operators and which are assumed to be unbounded. In the case when the operators C and B are relatively bounded with respect to the operators A and D, respectively, new conditions of the closeness or closability are obtained for the operator L. For the operator L acting in a Hilbert space the analogs of Rellich-Kato theorems on the stability of self-adjointness are obtained.

The spectroscopic line shape of electronic and vibrational excitations is ubiquitously described by a Fano profile. In the case of nearly symmetric and peaked Fano line shapes, the fit of the conventional Fano function to experimental data leads to difficulties in unambiguously extracting the asymmetry parameter, which may vary over orders of magnitude without degrading the quality of the fit. Moreover, the extracted asymmetry parameter depends on initially guessed values. Using the spectroscopic signature of the single-Co Kondo effect on Au(110) the ambiguity of the extracted asymmetry parameter is traced to the highly symmetric resonance profile combined with the inevitable scattering of experimental data. An improved parameterization of the conventional Fano function is suggested that enables the nonlinear optimization in a reduced parameter space. In addition, the presence of a global minimum in the sum of squared residuals and thus the independence of start parameters may conveniently be identified in a two-dimensional plot. An angular representation of the asymmetry parameter is suggested in order to reliably determine uncertainty margins via linear error propagation.