Javier MorenoMonika Winklmeier2026-03-222026-03-22202610.1017/s0013091525101272https://doi.org/10.1017/s0013091525101272https://andeanlibrary.org/handle/123456789/79156Abstract We consider a normal operator $T$ on a Hilbert space $H$ . Under various assumptions on the spectrum of $T$ , we give bounds for the spectrum of $T+A$ where $A$ is $T$ -bounded with relative bound less than 1 but we do not assume that $A$ is symmetric or normal. If the imaginary part of the spectrum of $T$ is bounded, then the spectrum of $T+A$ is contained in the region between two hyperbolas whose asymptotic slope depends on the $T$ -bound of $A$ . If the spectrum of $T$ is contained in a bisector, then the spectrum of $T+A$ is contained in the area between certain rotated hyperbolas. The case of infinitely many gaps in the spectrum of $T$ is studied. Moreover, we prove a stability result for the essential spectrum of $T+A$ . If $A$ is even $p$ -subordinate to $T$ , then we obtain stronger results for the localisation of the spectrum of $T+A$ .enSpectrum (functional analysis)MathematicsHilbert spaceOperator (biology)Operator matrixMathematical analysisDiscrete spectrumUpper and lower boundsHyperbolaSpace (punctuation)article