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this is an educational question.

can someone please explain with some simple examples:

(1) what is/are the spectrum of operator

(2) where it is useful

For providing examples of spectrum of operator, please also consider the following operator, matrix (A), differential operator, integral operator, .....

thank u.

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Dear Mozo, you've asked several somewhat poorly motivated questions in a short span of time that, in many ways, ask the answerer to do your work for you. Please be aware that this is generally bad etiquette to ask questions in this manner (especially since you have not accepted any answers yet). – Akhil Mathew Mar 28 '11 at 17:19
@Mathew. Dear Akhil, i think that one of the most motivational aspects of knowledge is to know its usefulness. And, that is my second question. As for as as accepting answer is concerned. I am still looking for a simple answer with simple examples. For example, spectrum of $d/dx$ and its usage. And, the most important of all I very much appreciate efforts of everyone who are trying to answer my questions. I do. – Mia Mar 28 '11 at 20:34

Since Wikipedia does already a good job explaining the definition of the spectrum, here is a short answer to (2):

Operators are complicated objects, so the basic strategy of introducing the spectrum of an operator is to associate a much simpler object, subsets of the complex plane, to a very complicated object, the operator, in such a way that it is possible to learn about properties of the operator by studying the spectrum.

In terms of differential and integral operators, the spectrum and its classification is about the question when a differential resp. integral equation has a solution, and what does go wrong if there is no solution.

In applications to physics, especially quantum physics, eigenvalues and their eigenvectors correspond to possible states of a given system.

On a more advanced level, symmetries of quantum physical systems are usually expressed by the invariance of physical states under transformations of representations of the symmetry group, where a representation of the symmetry group is a (strongly continuous) representation in the group of unital operators on a given Hilbert space. Properties of the physical system are encoded in properties of the spectral measure of the group representation, like, for example, the "mass gap" of the Yang-Mills millenium problem: In Minkowski spacetime the existence of a "mass gap" is equivalent with the property that the spectral measure of the translation group is bounded away from zero.

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I'll restrict myself to Hilbert spaces. The spectrum of an operator is the generalization of the concept of eigenvalues of a matrix. In other words a complex number $\lambda$ is in the spectrum of an operator $T$ if the operator $\lambda I - T$ is not invertible, where $I$ is the identity map.

So for a matrix this notion conincides with the eigenvalues of the matrix. For the differential operator it is a bit more involved and depends on the space the operator is acting on (compact or not for example). There is a good treatment of this in the book of Akhiezer and Glazman "Theory of Linear Operators in Hilbert Space".

The notion of spectrum allows one for example to define continuous maps of operators (selfadjoint or unitary for example). It is also used in solvind integral or differential equations.

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The spectrum of operator $A$ is the set of all complex numbers $z$ so that $zI-A$ is not invertible.

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