Abstract. We define a new type of spectrum, called the $\epsilon$-condition spectrum, of an element $a$ in a complex unital Banach algebra $A$ as $$\sigma_\epsilon(a):=\big\{\lambda\in{\msbm C} : \lambda-a \text{ is not invertible or } \|{\lambda-a}\|\|{(\lambda-a)^{-1}}\| \geq\frac{1}{\epsilon}\big\}. $$ This is expected to be useful in solving operator equations. We show that this is a particular case of the generalized spectrum defined by Ransford [10]. This $\epsilon$-condition spectrum shares some properties of the usual spectrum such as nonemptiness and compactness. But at the same time it has many properties that are different from the properties of the usual spectrum. For example, the $\epsilon$-condition spectrum always has only a finite number of components. Also if $a$ is not a scalar multiple of 1 then $\sigma_\epsilon(a)$ has no isolated points. Several examples are given to illustrate the main ideas.

AMS Subject Classification (1991): 46H05; 46J05

Keyword(s): Condition spectrum, Ransford spectrum, condition number

Received June 12, 2007, and in revised form September 21, 2007. (Registered under 6037/2009.)