The spectrum of a banach algebra is the set of all complex numbers for which the corresponding element in the algebra does not have an inverse. In other words, it is the set of all points at which the algebra is not invertible.
The spectrum of an individual element is a subset of the spectrum of the entire banach algebra. It consists of all points at which the element does not have an inverse within the algebra.
The spectrum is important because it provides information about the structure and properties of a banach algebra. It can be used to determine whether an element has an inverse, and can also be used to define important concepts such as the spectral radius and spectral mapping theorem.
The spectrum can be calculated using the Gelfand-Mazur theorem, which states that the spectrum of a commutative banach algebra is equal to the set of all nonzero homomorphisms from the algebra to the complex numbers. In practice, this means finding all possible linear functionals on the algebra and determining which ones are nonzero.
The spectrum has applications in a variety of mathematical fields, including functional analysis, operator theory, and differential equations. It is also used in physics and engineering to study systems with a continuous spectrum of eigenvalues. In addition, the spectrum plays a crucial role in the theory of C*-algebras, which are important in quantum mechanics.