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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.
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.