Spectrum of an operator

The spectrum of an operator is a set of special “values” that reveal important properties about how the operator acts on a space. Think of an operator as a complex system or function that transforms objects, and the spectrum includes those values where the operator behaves in a "special" or "resonant" way—like points where the system doesn't behave normally, or solutions that don’t exist in a usual sense. In mathematical terms, these are the values related to the operator's inability to be inverted or solved straightforwardly, providing insight into the operator’s fundamental characteristics.