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* 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.

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## Some Related Sentences

spectrum and any

An artist

**is****a**person engaged**in**one or more**of**__any__**of****a**broad__spectrum__**of**activities related to creating art**,**practicing**the**arts and / or demonstrating an art**.**
Black

**is****the**color**of**objects that do not emit or reflect light**in**__any__part**of****the**visible__spectrum__; they absorb all such frequencies**of**light**.****The**

__spectrum__

**of**

__any__element x

**is**

**a**closed

**subset**

**of**

**the**closed disc

**in**

**C**with radius || x || and center 0

**,**and thus

**is**

**compact**

**.**

For instance

**,****a**diagonal**operator****on****the**Hilbert**space**may have__any__**compact****nonempty****subset****of****C****as**__spectrum__**.*******

**The**

__spectrum__

**of**

__any__commutative ring with

**the**Zariski topology ( that

**is**

**,**

**the**set

**of**all prime ideals )

**is**

**compact**

**,**but never Hausdorff ( except

**in**trivial cases ).

Spectroscopic studies revealed absorption lines

**in****the**Jovian__spectrum__due to diatomic sulfur ( S < sub > 2 </ sub >) and carbon disulfide ( CS < sub > 2 </ sub >),**the**first detection**of**either**in**Jupiter**,**and only**the**second detection**of**S < sub > 2 </ sub >**in**__any__astronomical object**.**
These are capable

**of****the**most severe types**of**molecular damage**,**which can happen**in**biology to__any__type**of**biomolecule**,**including mutation and cancer**,**and often at great depths from**the**skin**,**since**the**higher end**of****the**X-ray__spectrum__**,**and all**of****the**gamma ray__spectrum__**,**are penetrating to matter**.**
Fascists have commonly opposed having

**a**firm association with__any__section**of****the**left-right__spectrum__**,**considering it inadequate to describe their beliefs**,**though fascism's goal to promote**the**rule**of**people deemed innately superior while seeking to purge society**of**people deemed innately inferior**is**identified**as****a**prominent far-right theme**.**
Individuals were categorized according to their so-called " rejection

__spectrum__" which allowed doctors to counter__any__immune system responses to**the**new organs**,**allowing transplants to " take " for life**.**
We can also show that

**the**possible values**of****the**observable A**in**__any__state must belong to**the**__spectrum__**of**A**.**
In linguistics

**the**term orthography**is**often used to refer to__any__method**of**writing**a**language**,**without judgment**as**to right and wrong**,**with**a**scientific understanding that orthographic standardization exists**on****a**__spectrum__**of**strength**of**convention**.**
Pacifism covers

**a**__spectrum__**of**views ranging from**the**belief that international disputes can and should be peacefully resolved ; to calls for**the**abolition**of****the**institutions**of****the**military and war ; to opposition to__any__organization**of**society through governmental force ( anarchist or libertarian pacifism ); to rejection**of****the**use**of**physical violence to obtain political**,**economic or social goals ; to opposition to violence under__any__circumstance**,**including defense**of**self and others**.**
Pacifism covers

**a**__spectrum__**of**views**,**including**the**belief that international disputes can and should be peacefully resolved**,**calls for**the**abolition**of****the**institutions**of****the**military and war**,**opposition to__any__organization**of**society through governmental force ( anarchist or libertarian pacifism ), rejection**of****the**use**of**physical violence to obtain political**,**economic or social goals**,****the**obliteration**of**force except**in**cases where it**is**absolutely necessary to advance**the**cause**of**peace**,**and opposition to violence under__any__circumstance**,**even defense**of**self and others**.**
If

**the**surface has__any__transparent or translucent properties**,**it refracts**a**portion**of****the**light beam into itself**in****a**different direction while absorbing**some**( or all )**of****the**__spectrum__( and possibly altering**the**color ).
Fundamental strings exist

**in**9 dimensions and**the**strings can vibrate**in**__any__direction**,**meaning that**the**__spectrum__**of**vibrational modes**is**much richer**.**
As with

__any__religious movement**,****a**theological__spectrum__exists within Adventism comparable to**the**fundamentalist-moderate-liberal__spectrum__**in****the**wider Christian church and**in**other religions**.**
In general

**,**__any__particular instrument will operate over**a**small portion**of****this**total range because**of****the**different techniques used to measure different portions**of****the**__spectrum__**.**

spectrum and bounded

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__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

**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__

**linear**

**operator**T acting

**on**

**a**

**Banach**

**space**X

**is**

**the**set

**of**

**complex**

**numbers**λ such that λI − T does not have an inverse that

**is**

**a**

__bounded__

**linear**

**operator**

**.**

**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__**.**
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**

**.**

spectrum and linear

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**.**1.317 seconds.