On some results on condition spectrum

Condition spectrum of an element $a$ in a Banach algebra 𝒜 is studied. Its relationships with the spectrum and numerical range are given which provides an upper bound and lower bound of the condition spectrum as well. It is demonstrated that for G 1 —class elements, the derived subset equals the condition spectrum. Suitable examples are given for illustration. Characterization of Hermitian and scalar elements by means of their condition spectrum are given. Furthermore, we have demonstrated that in the case of Toeplitz operators and matrices, the enclosures obtained through the upper bound and lower bound improve upon previous estimates of the bounds in [7].