Spectral theorem

The term spectral theorem refers to various related mathematical statements from linear algebra and functional analysis . The simplest variant makes a statement about the diagonalisability of a certain class of matrices . The other spectral theorems considered here transfer this principle to operators between infinite-dimensional spaces. The name is derived from the "spectrum" of eigenvalues .

Spectral theorem for endomorphisms of finite-dimensional vector spaces

statement

For a finite-dimensional unitary vector space ( or ) there exists an orthonormal basis of eigenvectors of an endomorphism if and only if this is normal and all eigenvalues belong. ${\ displaystyle \ mathbb {K}}$${\ displaystyle \ mathbb {K} = \ mathbb {R}}$${\ displaystyle \ mathbb {K} = \ mathbb {C}}$${\ displaystyle \ mathbb {K}}$

In matrix terms this means that a matrix is ​​unitary diagonalizable if and only if it is normal and only has eigenvalues ​​out . Another common formulation is that a matrix is normal if and only if it can be unitarily diagonalized, i.e. a unitary matrix (of the same dimension) exists such that ${\ displaystyle \ mathbb {K}}$${\ displaystyle A}$ ${\ displaystyle U}$

${\ displaystyle U ^ {*} AU = D}$

with , is a diagonal matrix with the eigenvalues of on the main diagonal . ${\ displaystyle D: = {\ text {diag}} (\ lambda _ {1}, \ ldots, \ lambda _ {n})}$${\ displaystyle \ lambda _ {1}, \ ldots, \ lambda _ {n}}$${\ displaystyle A}$

Remarks

• For the condition that all eigenvalues lie in is always fulfilled ( is algebraically closed according to the fundamental theorem of algebra ), so here all normal matrices are unitary diagonalizable. For this does not apply.${\ displaystyle \ mathbb {K} = \ mathbb {C}}$${\ displaystyle \ mathbb {K}}$${\ displaystyle \ mathbb {C}}$${\ displaystyle \ mathbb {K} = \ mathbb {R}}$
• A self-adjoint endomorphism or a Hermitian matrix only has real eigenvalues. The spectral theorem says that all Hermitian matrices are diagonalizable and that an endomorphism is self-adjoint if and only if there is an orthonormal basis of eigenvectors and all eigenvalues ​​are real. In particular, real symmetric matrices are always diagonalizable.

Spectral theorem for compact operators

statement

Let a - Hilbert space and a linear compact operator , which is normal in the case and self-adjoint in the case . Then there exists a (possibly finite) orthonormal system and a null sequence in such that ${\ displaystyle H}$${\ displaystyle \ mathbb {K}}$${\ displaystyle T \ colon H \ to H}$${\ displaystyle \ mathbb {K} = \ mathbb {C}}$ ${\ displaystyle \ mathbb {K} = \ mathbb {R}}$ ${\ displaystyle e_ {1}, e_ {2}, \ ldots}$ ${\ displaystyle (\ lambda _ {k}) _ {k \ in \ mathbb {N}}}$${\ displaystyle \ mathbb {K} \ backslash \ {0 \}}$

${\ displaystyle H = \ ker (T) \ oplus {\ overline {\ operatorname {span} (\ {e_ {1}, e_ {2}, \ ldots \})}}}$

such as

${\ displaystyle Tx = \ sum _ {k = 1} ^ {\ infty} \ lambda _ {k} \ langle e_ {k}, x \ rangle e_ {k}}$

applies to all . They are for all eigenvalues of and is an eigenvector to . In addition , where is the operator norm. ${\ displaystyle x \ in H}$${\ displaystyle \ lambda _ {k}}$${\ displaystyle k \ in \ mathbb {N}}$ ${\ displaystyle T}$${\ displaystyle e_ {k}}$${\ displaystyle \ lambda _ {k}}$${\ displaystyle \ textstyle \ | T \ | = \ sup _ {k \ in \ mathbb {N}} | \ lambda _ {k} |}$${\ displaystyle \ | \ cdot \ |}$

Projection version of the spectral set

One can reformulate the spectral theorem for compact operators with the help of orthogonal projections . Let again be a -Hilbert space and a linear compact operator which is normal in the case and self-adjoint in the case . The orthogonal projection onto the eigenspace that belongs to is referred to. The operator thus has the representation , where is the dimension of the eigenspace and an orthonormal basis of the eigenspace. Then the spectral theorem can be reformulated: There is a null sequence of eigenvalues in such that ${\ displaystyle H}$${\ displaystyle \ mathbb {K}}$${\ displaystyle T \ colon H \ to H}$${\ displaystyle \ mathbb {K} = \ mathbb {C}}$${\ displaystyle \ mathbb {K} = \ mathbb {R}}$${\ displaystyle E_ {k}}$${\ displaystyle \ lambda _ {k}}$ ${\ displaystyle \ operatorname {ker} (\ lambda _ {k} \ mathrm {id} _ {H} -T)}$${\ displaystyle E_ {k}}$${\ displaystyle \ textstyle E_ {k} x = \ sum _ {i = 1} ^ {d_ {k}} \ langle e_ {i} ^ {k}, x \ rangle e_ {i} ^ {k}}$${\ displaystyle d_ {k}}$${\ displaystyle \ operatorname {ker} (\ lambda _ {k} \ mathrm {id} _ {H} -T)}$${\ displaystyle \ {e_ {1} ^ {k}, \ ldots, e_ {d_ {k}} ^ {k} \}}$${\ displaystyle (\ lambda _ {k}) _ {k \ in \ mathbb {N}}}$${\ displaystyle \ mathbb {K} \ backslash \ {0 \}}$

${\ displaystyle Tx = \ sum _ {k = 1} ^ {\ infty} \ lambda _ {k} E_ {k} x}$

applies to all . This series converges not only pointwise but also with respect to the operator norm. ${\ displaystyle x \ in H}$

Spectral theorem for bounded operators

statement

Let be a Hilbert space and a self-adjoint continuous linear operator . Then there is a clearly determined spectral dimension with a compact carrier in with ${\ displaystyle H}$${\ displaystyle T \ colon H \ to H}$ ${\ displaystyle E \ colon \ Sigma \ to L (H, H)}$${\ displaystyle \ mathbb {R}}$

${\ displaystyle T = \ int _ {\ sigma (T)} \ lambda \, \ mathrm {d} E _ {\ lambda}.}$

Here called the Borel σ-algebra of , the set of bounded operators on and the range of . ${\ displaystyle \ Sigma}$${\ displaystyle \ mathbb {R}}$${\ displaystyle L (H, H)}$${\ displaystyle H}$${\ displaystyle \ sigma (T)}$${\ displaystyle T}$

Relation to the previous spectral sentences

• If finite-dimensional, so it holds, then the self-adjoint operator has the pairwise different eigenvalues and it holds, as already shown in the article, where the orthogonal projection onto the eigenspace of is. The spectral measure of is then given for all by . Therefore the spectral theorem for bounded operators is reduced to the spectral theorem from linear algebra.${\ displaystyle H}$${\ displaystyle H \ cong \ mathbb {C} ^ {n}}$${\ displaystyle T}$${\ displaystyle \ mu _ {1}, \ ldots, \ mu _ {m}}$
${\ displaystyle T = \ sum _ {i = 1} ^ {m} \ mu _ {i} E _ {\ {\ mu _ {i} \}},}$
${\ displaystyle E _ {\ {\ mu _ {i} \}}}$${\ displaystyle \ operatorname {ker} (\ mu _ {i} -T)}$${\ displaystyle \ mu _ {i}}$${\ displaystyle T}$${\ displaystyle A \ in \ Sigma}$
${\ displaystyle E_ {A} = \ sum _ {\ {i: \ mu _ {i} \ in A \}} E _ {\ {\ mu _ {i} \}}}$
${\ displaystyle \ textstyle T = \ sum _ {i = 1} ^ {m} \ mu _ {i} E _ {\ {\ mu _ {i} \}}}$
• Let it be a linear compact operator, so it was also shown in the article that a spectral theorem exists for such operators. If the sequence of eigenvalues ​​of and one again chooses as the spectral measure , whereby the sum then generally has countably many summands and converges point by point, but not with respect to the operator norm , then the spectral theorem for bounded operators is simplified to Therefore, the spectral theorem for bounded operators includes also the spectral theorem for compact operators.${\ displaystyle T \ colon H \ to H}$${\ displaystyle (\ mu _ {i}) _ {i \ in \ mathbb {N}}}$${\ displaystyle T}$${\ displaystyle \ textstyle E_ {A} = \ sum _ {\ {i: \ mu _ {i} \ in A \}} E _ {\ {\ mu _ {i} \}}}$
${\ displaystyle T = \ sum _ {i = 1} ^ {\ infty} \ mu _ {i} E _ {\ {\ mu _ {i} \}}.}$

example

The operator defined by is self-adjoint with and has no eigenvalues. The spectral measure with is a spectral measure with a compact carrier. It represents because it applies ${\ displaystyle T \ colon L ^ {2} ([0,1]) \ to L ^ {2} ([0,1])}$${\ displaystyle T (x) (t) = t \ cdot x (t)}$${\ displaystyle \ sigma (T) \ subset [0,1]}$${\ displaystyle E_ {A} x = \ chi _ {A \ cap [0,1]} x}$${\ displaystyle A \ in \ Sigma}$${\ displaystyle T}$

${\ displaystyle \ int \ lambda \, \ mathrm {d} \ langle E _ {\ lambda} x, y \ rangle = \ int _ {[0,1]} \ lambda x (\ lambda) {\ overline {y ( \ lambda)}} \, \ mathrm {d} \ lambda = \ langle Tx, y \ rangle _ {L ^ {2} ([0,1])}.}$

Measurable functional calculus

Be a self-adjoint operator. The measurable functional calculus is a uniquely determined, continuous, involutive algebra homomorphism . With the help of the spectral decomposition one gets a simple representation of this figure. It is true ${\ displaystyle T \ in L (H, H)}$${\ displaystyle {\ hat {\ Phi}} \ colon {\ mathcal {B}} (\ sigma (T)) \ to L (H, H)}$

${\ displaystyle {\ hat {\ Phi}} (f) = f (T) = \ int _ {\ sigma (T)} f (\ lambda) \ mathrm {d} E _ {\ lambda}.}$

Spectral theorem for unlimited operators

If there is a tightly defined normal operator on a complex Hilbert space , there is a clearly determined spectral measure on the Borel sets of , so that the following applies ( be the spectrum of ): ${\ displaystyle A}$ ${\ displaystyle H}$${\ displaystyle E}$${\ displaystyle \ mathbb {C}}$${\ displaystyle \ sigma (A)}$${\ displaystyle A}$

• ${\ displaystyle A = \ int _ {z \ in \ sigma (A)} z \, \ mathrm {d} E (z)}$
• For a lot with is true .${\ displaystyle M \ subseteq \ mathbb {C}}$${\ displaystyle M \ cap \ sigma (A) = \ emptyset}$${\ displaystyle E (M) = 0}$
• For an open set with applies .${\ displaystyle M \ subseteq \ mathbb {C}}$${\ displaystyle M \ cap \ sigma (A) \ neq \ emptyset}$${\ displaystyle E (M) \ neq 0}$

A self-adjoint operator is normal with a real spectrum; one can restrict the above integral to real numbers.

The domain of definition is given by

${\ displaystyle D (A) = \ left \ {x \ in H \ left | \ int _ {\ sigma (A)} | \ lambda | ^ {2} \ mathrm {d} \ langle E _ {\ lambda} x , x \ rangle <\ infty \ right. \ right \}}$

and the square shape area

${\ displaystyle Q (A) = \ left \ {x \ in H \ left | \ int _ {\ sigma (A)} | \ lambda | \ mathrm {d} \ langle E _ {\ lambda} x, x \ rangle <\ infty \ right. \ right \}}$.

The latter is obviously the maximum domain for the associated square shape, which is particularly important in quantum mechanics. ${\ displaystyle \ langle Ax, x \ rangle}$

An equivalent formulation of the spectral theorem is that unitary is equivalent to a multiplication operator over a space (for a measure space ) with a complex-valued measurable function ; is self adjoint, then is real. ${\ displaystyle A}$ ${\ displaystyle L_ {2} (\ Omega)}$ ${\ displaystyle \ Omega}$ ${\ displaystyle f \ colon \ Omega \ to \ mathbb {C}}$${\ displaystyle A}$${\ displaystyle f}$

A normal operator in the complex can usually be written as the sum of two interchangeable self-adjoint operators multiplied by the real or the imaginary unit ( "real part" + "imaginary part"${\ displaystyle i \,}$ ). Furthermore, due to the interchangeability of the - the operator and the operator have the same eigenvectors (despite possibly different eigenvalues ). So could be a function of the self adjoint operator , with appropriate . Then ultimately only a single (real!) Spectral representation would be important, for example that of , and it would apply, for example, that   and   is. ${\ displaystyle A = {\ hat {W}} _ {1} + i {\ hat {W}} _ {2} \ ,, {\ hat {W}} _ {i} \ equiv {\ hat {W }} _ {i} ^ {\ dagger} \ ,, \, {\ hat {W}} _ {1} {\ hat {W}} _ {2} = {\ hat {W}} _ {2} {\ hat {W}} _ {1} \ ,.}$${\ displaystyle {\ hat {W}} _ {i}}$${\ displaystyle {\ hat {W}} _ {2} \,}$${\ displaystyle {\ hat {W}} _ {1}}$${\ displaystyle W_ {2}}$${\ displaystyle {\ hat {W}} _ {1}}$${\ displaystyle {\ hat {W}} _ {2} \ equiv f_ {2} ({\ hat {W}} _ {1}) \ ,,}$${\ displaystyle f_ {2}}$${\ displaystyle {\ hat {W}} _ {1} \, \, (= {\ frac {A + A ^ {\ dagger}} {2}})}$
${\ displaystyle \ textstyle {\ hat {W}} _ {1} = \ int _ {x \ in \ sigma ({\ hat {W}} _ {1})} \, x \, \ mathrm {d} Ex)}$${\ displaystyle \ textstyle {\ hat {W}} _ {2} \, \, (= {\ frac {AA ^ {\ dagger}} {2i}}) = \ int _ {x \ in \ sigma ({ \ hat {W}} _ {1})} \, f_ {2} (x) \, \ mathrm {d} E (x)}$

Role in quantum mechanics

In quantum mechanics , the spectral theorem ("evolution theorem") is of central importance, since measurable physical quantities, so-called " observables ", are represented by self-adjoint operators on a Hilbert space.

The possible measured values ​​of an observable correspond to its spectrum, which is divided into a point spectrum (or “discrete spectrum”) and a continuous spectrum. The elements of the point spectrum are also called eigenvalues. For a discrete observable, i. H. an observable without a continuous spectrum, is the probability of obtaining the measured value for a given quantum mechanical state , given by the square of the magnitude of the scalar product , where the eigenfunction is the eigenvalue . ${\ displaystyle | \ psi \ rangle}$${\ displaystyle \ lambda _ {j}}$${\ displaystyle \ langle \ phi _ {j} | \ psi \ rangle}$${\ displaystyle \ phi _ {j}}$${\ displaystyle \ lambda _ {j}}$

history

The spectral theorem for compact self-adjoint operators and that for bounded self-adjoint operators go back in particular to the work of David Hilbert . In 1906, in his 4th communication, he published evidence of these statements. Hilbert's presentation of the sentences is of course very different from today's presentation. Instead of the spectral measure, he used the Stieltjes integral , which Thomas Jean Stieltjes had only introduced in 1894 to investigate continued fractions . According to Hilbert, proofs were found for the spectral theorem for bounded and unrestricted operators by, among others, Riesz (1930-1932) and Lengyel and Stone (1936) and for the unrestricted case also by Leinfelder (1979).