When the Banach algebra A is the algebra L ( X ) of bounded linear operators on a complex Banach space X ( e. g., the algebra of square matrices ), the notion of the spectrum in A coincides with the usual one in the operator theory.

* The spectrum of any bounded linear operator on a Banach space is a nonempty compact subset of the complex numbers C. Conversely, any compact subset of C arises in this manner, as the spectrum of some bounded linear operator.

The spectrum of any bounded symmetric operator is real ; in particular all its eigenvalues are real, although a symmetric operator may have no eigenvalues.

In functional analysis, the concept of the spectrum of a bounded operator is a generalisation of the concept of eigenvalues for matrices.

Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the identity operator.

In fact every bounded linear operator on a complex Banach space must have a non-empty spectrum.

In this case a complex number λ is said to be in the spectrum of such an operator T: D → X ( where D is dense in X ) if there is no bounded inverse ( λI − T )< sup >− 1 </ sup >: X → D.

The spectrum of a bounded operator T is always a closed, bounded and non-empty subset of the complex plane.

The boundedness of the spectrum follows from the Neumann series expansion in λ ; the spectrum σ ( T ) is bounded by || T ||.

So the spectrum includes the set of approximate eigenvalues, which are those λ such that is not bounded below ; equivalently, it is the set of λ for which there is a sequence of unit vectors x < sub > 1 </ sub >, x < sub > 2 </ sub >, ... for which

It can be shown that, in general, the approximate point spectrum of a bounded multiplication operator is its spectrum.

One can classify the spectrum in exactly the same way as in the bounded case.

For λ to be in the resolvent ( i. e. not in the spectrum ), as in the bounded case λI − T must be bijective, since it must have a two-sided inverse.

Therefore, as in the bounded case, a complex number λ lies in the spectrum of a closed operator T if and only if λI − T is not bijective.

* Positive element of a C *- algebra ( such as a bounded linear operator ) whose spectrum consists of positive real numbers

In mathematics, the spectral radius of a square matrix or a bounded linear operator is the supremum among the absolute values of the elements in its spectrum, which is sometimes denoted by ρ (·).

The Koszul complex is essential in defining the joint spectrum of a tuple of bounded linear operators in a Banach space.

Codecs like FLAC, Shorten and TTA use linear prediction to estimate the spectrum of the signal.

LPC may also be thought of as a basic perceptual coding technique ; reconstruction of an audio signal using a linear predictor shapes the coder's quantization noise into the spectrum of the target signal, partially masking it.

In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues ; more precisely: as spectral values ( point spectrum plus absolute continuous plus singular continuous spectrum ) of linear operators in Hilbert space.

In linear prediction coding, the all-pole filter replaces the bandpass filter bank of its predecessor and is used at the encoder to whiten the signal ( i. e., flatten the spectrum ) and again at the decoder to re-apply the spectral shape of the target speech signal.

Categorizing males and females into social roles creates binaries in which individuals feel they have to be at one end of a linear spectrum and must identify themselves as man or woman.

Pink noise ( left ) and white noise ( right ) on an FFT spectrogram with linear frequency vertical axis ( on a typical audio or similar spectrum analyzer the pink noise would be flat, not downward-sloping, and the white noise rising )

FLASH ( Free-electron-LASer in Hamburg ) is a superconducting linear accelerator with a free electron laser for radiation in the vacuum-ultraviolet and soft X-ray range of the spectrum.

One prime example is the invention of the linear phase equalizer, which has inherent phase shift that is homogeneous across the frequency spectrum.

In the ideal case of a single sharp energy barrier for the tip-sample interactions the dynamic force spectrum will show a linear increase of the rupture force as function of a logarithm of the loading rate.

A complex number λ is said to be in the resolvent set, that is, the complement of the spectrum of a linear operator

* Spec k, the spectrum of the polynomial ring over a field k, which is also denoted, the affine line: the polynomial ring is known to be a principal ideal domain and the irreducible polynomials are the prime elements of k. If k is algebraically closed, for example the field of complex numbers, a non-constant polynomial is irreducible if and only if it is linear, of the form t − a, for some element a of k. So, the spectrum consists of one closed point for every element a of k and a generic point, corresponding to the zero ideal.

In sound processing, the mel-frequency cepstrum ( MFC ) is a representation of the short-term power spectrum of a sound, based on a linear cosine transform of a log power spectrum on a nonlinear mel scale of frequency.

Once the operator took his finger off, the player would speed up until its tachometer was back in phase with the master, and as this happened, the phasing effect would appear to slide up the frequency spectrum.

For instance, a diagonal operator on the Hilbert space may have any compact nonempty subset of C as spectrum.

* Spectrum of an operator, in functional analysis ( a generalisation of the spectrum of a matrix )

Once the operator took his finger off, the player would speed up until its tachometer was back in phase with the master, and as this happened, the phasing effect would appear to slide up the frequency spectrum.

In simpler instruments the absorption is determined one wavelength at a time and then compiled into a spectrum by the operator.

One of the key results of loop quantum gravity is quantization of areas: the operator of the area of a two-dimensional surface should have a discrete spectrum.

In the absence of such control or alternative arrangements such as a privatized electromagnetic spectrum, chaos might result if, for example, airlines didn't have specific frequencies to work under and an amateur radio operator were interfering with the pilot's ability to land an aircraft.

Contrary to what is sometimes claimed in introductory physics textbooks, it is possible for symmetric operators to have no eigenvalues at all ( although the spectrum of any self-adjoint operator is nonempty ).

Residual spectrum of a normal operator is empty.

This notion can be extended to the spectrum of an operator in the infinite-dimensional case.

The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues.

However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues.

On the other hand 0 is in the spectrum because the operator R − 0 ( i. e. R itself ) is not invertible: it is not surjective since any vector with non-zero first component is not in its range.