In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number is said to be in the spectrum of a bounded linear operator if is not invertible, where is the identity operator. The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics. 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. For example, consider the right shift operator R on the Hilbert space ℓ2, This has no eigenvalues, since if Rx=λx then by expanding this expression we see that x1=0, x2=0, etc. 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. In fact every bounded linear operator on a complex Banach space must have a non-empty spectrum. The notion of spectrum extends to unbounded (i.e. not necessarily bounded) operators. A complex number λ is said to be in the spectrum of an unbounded operator defined on domain if there is no bounded inverse defined on the whole of If T is closed (which includes the case when T is bounded), boundedness of follows automatically from its existence. The space of bounded linear operators B(X) on a Banach space X is an example of a unital Banach algebra. Since the definition of the spectrum does not mention any properties of B(X) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim. (Wikipedia).
Spectral Theory 6 - Spectrum of Compact Operators (Functional Analysis - Part 33)
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From playlist Functional analysis
Spectral Theory 1 - Spectrum of Bounded Operators (Functional Analysis - Part 28)
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From playlist Functional analysis
Spectral Theory 3 - Properties of the spectrum (Functional Analysis - Part 30)
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From playlist Functional analysis
Spectral Theory 7 - Spectral Theorem for Compact Operators (Functional Analysis - Part 34)
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From playlist Functional analysis
Spectral Theory 2 - Spectrum of Multiplication Operator (Functional Analysis - Part 29)
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From playlist Functional analysis
Spectral Theory 5 - Normal and Self-Adjoint Operators (Functional Analysis - Part 32)
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From playlist Functional analysis
Determine if the equation represents a function
👉 Learn how to determine whether relations such as equations, graphs, ordered pairs, mapping and tables represent a function. A function is defined as a rule which assigns an input to a unique output. Hence, one major requirement of a function is that the function yields one and only one r
From playlist What is the Domain and Range of the Function
In this video, I talk about the definition of a function and properties of functions. I also go over some examples of how to determine whether a relation is a function or not and how to evaluate functions. Enjoy! Facebook: https://www.facebook.com/braingainzofficial Instagram: https://
From playlist College Algebra
Stefano Baroni - estimate transport coefficients from short equilibrium molecular-dynamic simulation
Recorded 06 May 2022. Stefano Baroni of the International School for Advanced Studies presents "Separating flour from bran: how to optimally estimate transport coefficients from short equilibrium molecular-dynamics simulations" at IPAM's Large-Scale Certified Numerical Methods in Quantum M
From playlist 2022 Large-Scale Certified Numerical Methods in Quantum Mechanics
Henryk Iwaniec, Spectral Theory of Automorphic Forms and Analytic Number Theory [2001]
Slides for this talk: https://drive.google.com/file/d/1EDyLbE9Aqk_61njU26gbYoHoR93aetAq/view?usp=sharing Henryk Iwaniec (Rutgers Univ) Spectral Theory of Automorphic Forms and Analytic Number Theory WEDNESDAY, APRIL 4, 2001 11:00 - 12:00 Conference on Automorphic Forms: Concepts, Te
From playlist Number Theory
Rainer von Sachs: Time-frequency analysis of locally stationary Hawkes processes
Abstract : In this talk we address generalisation of stationary Hawkes processes in order to allow for a time-evolutive second-order analysis. A formal derivation of a time-frequency analysis via a time-varying Bartlett spectrum is given by introduction of the new class of locally stationa
From playlist Probability and Statistics
André Voros - Resurgent Theta-functions...
Resurgent Theta-functions: a conjectured gateway into dimension D superior at 1 quantum mechanics Resurgent analysis of the stationary Schrödinger equation (exact-WKB method) has remained exclusivelyconfined to 1D systems due to its underlying linear-ODE techniques.Here, b
From playlist Resurgence in Mathematics and Physics
Koopman Spectral Analysis (Continuous Spectrum)
In this video, we discuss how to use Koopman theory for dynamical systems with a continuous eigenvalue spectrum. These systems are quite common, such as a pendulum, where the period deforms continuously as energy is added to the system. To handle these systems, we use a neural network a
From playlist Koopman Analysis
On the Possibility of Primordial Features by Dhiraj Hazra
PROGRAM: PHYSICS OF THE EARLY UNIVERSE - AN ONLINE PRECURSOR ORGANIZERS: Robert Brandenberger (McGill University, Montreal, Canada), Jerome Martin (Institut d'Astrophysique de Paris, France), Subodh Patil (Instituut-Lorentz for Theoretical Physics, Leiden, Netherlands) and L Sriramkumar (
From playlist Physics of The Early Universe - An Online Precursor
Using the vertical line test to determine if a graph is a function or not
👉 Learn how to determine whether relations such as equations, graphs, ordered pairs, mapping and tables represent a function. A function is defined as a rule which assigns an input to a unique output. Hence, one major requirement of a function is that the function yields one and only one r
From playlist What is the Domain and Range of the Function
Neuroscience source separation 1b: Spectral separation in MATLAB
This is part one of a three-part lecture series I taught in a masters-level neuroscience course in fall of 2020 at the Donders Institute (the Netherlands). The lectures were all online in order to minimize the spread of the coronavirus. That's good for you, because now you can watch the en
From playlist Neuroscience source separation (3-part lecture series)
Lecture 15, Discrete-Time Modulation | MIT RES.6.007 Signals and Systems, Spring 2011
Lecture 15, Discrete-Time Modulation Instructor: Alan V. Oppenheim View the complete course: http://ocw.mit.edu/RES-6.007S11 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT RES.6.007 Signals and Systems, 1987
How to do the 3x2pt analysis of SZ and galaxies - Komatsu - Workshop 2 - CEB T3 2018
Eiichiro Komatsu (Max Planck Institute for Astrophysics) / 25.10.2018 How to do the 3x2pt analysis of SZ and galaxies ---------------------------------- Vous pouvez nous rejoindre sur les réseaux sociaux pour suivre nos actualités. Facebook : https://www.facebook.com/InstitutHenriPoinc
From playlist 2018 - T3 - Analytics, Inference, and Computation in Cosmology
Functions of equations - IS IT A FUNCTION
👉 Learn how to determine whether relations such as equations, graphs, ordered pairs, mapping and tables represent a function. A function is defined as a rule which assigns an input to a unique output. Hence, one major requirement of a function is that the function yields one and only one r
From playlist What is the Domain and Range of the Function