Unlike the simple case of a matrix, in infinite dimensional vector spaces, the spectrum may be continuous.
The quintessential example of that is the spectrum of the position operator in quantum mechanics, in which any real number is a possible eigenvalue, since the particle may be found in any position. The associated eigenvectors are the corresponding Dirac delta functions.
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The spectrum of a linear operator consists of the scalars for which is not invertible. In finite dimensions it is the set of eigenvalues.
In functional analysis, the notion of the spectrum of an operator is a fundamental concept that extends the idea of eigenvalues from finite-dimensional linear algebra to more general settings, particularly in the study of bounded linear operators on Banach spaces and Hilbert spaces.