##  [Spectrum](/spectrum) 

  ##  [Spectrum](https://puremath.quantumdictionary.io/spectrum-0) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The set of scalars λ for which the operator shift T−λI fails to be invertible; it captures the total spectral properties of an operator, generalizing eigenvalues to infinite-dimensional and non-normal contexts.

 

 

 

 

 





 

 



 ##  [Spectrum](https://algebra.quantumdictionary.io/spectrum-1) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The set of scalars λ in the base field (typically C) for which the operator T−λI fails to be invertible in the ambient algebra/topology; for bounded operators on Banach spaces the spectrum splits into point, continuous and residual parts.