operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operato ...

, quasinormal operators is a class of

bounded operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector ...

s defined by weakening the requirements of a

normal operator In mathematics, especially functional analysis, a normal operator on a complex number, complex Hilbert space H is a continuous function (topology), continuous linear operator N\colon H\rightarrow H that commutator, commutes with its Hermitian adjo ...

. Every quasinormal operator is a

subnormal operator In mathematics, especially operator theory, subnormal operators are bounded operators on a Hilbert space defined by weakening the requirements for normal operators. Some examples of subnormal operators are isometry, isometries and Toeplitz operator ...

. Every quasinormal operator on a finite-dimensional

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

is normal.

Definition and some properties

Definition

Let ''A'' be a bounded operator on a Hilbert space ''H'', then ''A'' is said to be quasinormal if ''A'' commutes with ''A*A'', i.e. :

A(A^*A) = (A^*A) A.\,

Properties

A normal operator is necessarily quasinormal. Let ''A'' = ''UP'' be the

polar decomposition In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is a unitary matrix, and P is a positive semi-definite Hermitian matrix (U is an orthogonal matrix, and P is a posit ...

of ''A''. If ''A'' is quasinormal, then ''UP = PU''. To see this, notice that the positive factor ''P'' in the polar decomposition is of the form (''A*A''), the unique positive square root of ''A*A''. Quasinormality means ''A'' commutes with ''A*A''. As a consequence of the

continuous functional calculus In mathematics, particularly in operator theory and C*-algebra theory, the continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra. In advanced theory, the ap ...

for

self-adjoint operator In mathematics, a self-adjoint operator on a complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. That is, \langle Ax,y \rangle = \langle x,Ay \rangle for al ...

s, ''A'' commutes with ''P'' = (''A*A'') also, i.e. :

U P P = P U P.\,

So ''UP = PU'' on the range of ''P''. On the other hand, if ''h'' ∈ ''H'' lies in kernel of ''P'', clearly ''UP h'' = 0. But ''PU h'' = 0 as well. because ''U'' is a

partial isometry Partial may refer to: Mathematics *Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant ** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial ...

whose initial space is closure of range ''P''. Finally, the self-adjointness of ''P'' implies that ''H'' is the direct sum of its range and kernel. Thus the argument given proves ''UP'' = ''PU'' on all of ''H''. On the other hand, one can readily verify that if ''UP'' = ''PU'', then ''A'' must be quasinormal. Thus the operator ''A'' is quasinormal if and only if ''UP'' = ''PU''. When ''H'' is finite dimensional, every quasinormal operator ''A'' is normal. This is because that in the finite dimensional case, the partial isometry ''U'' in the polar decomposition ''A'' = ''UP'' can be taken to be unitary. This then gives :

A^*A = (UP)^* UP =  PU (PU)^* = AA^*.\,

In general, a partial isometry may not be extendable to a unitary operator and therefore a quasinormal operator need not be normal. For example, consider the

unilateral shift In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function to its translation . In time series analysis, the shift operator is called the '' lag opera ...

''T''. ''T'' is quasinormal because ''T*T'' is the identity operator. But ''T'' is clearly not normal.

Quasinormal invariant subspaces

It is not known that, in general, whether a bounded operator ''A'' on a Hilbert space ''H'' has a nontrivial invariant subspace. However, when ''A'' is normal, an affirmative answer is given by the

spectral theorem In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involvin ...

. Every normal operator ''A'' is obtained by integrating the identity function with respect to a spectral measure ''E'' = on the spectrum of ''A'', ''σ''(''A''): :

A = \int_ \lambda \, dE (\lambda).\,

For any Borel set ''B'' ⊂ ''σ''(''A''), the projection ''E_B'' commutes with ''A'' and therefore the range of ''E_B'' is an invariant subspace of ''A''. The above can be extended directly to quasinormal operators. To say ''A'' commutes with ''A*A'' is to say that ''A'' commutes with (''A*A''). But this implies that ''A'' commutes with any projection ''E_B'' in the spectral measure of (''A*A'')^{{{frac, 1, 2}, which proves the invariant subspace claim. In fact, one can conclude something stronger. The range of ''E_B'' is actually a ''

reducing subspace Reduction, reduced, or reduce may refer to: Science and technology Chemistry * Reduction (chemistry), part of a reduction-oxidation (redox) reaction in which atoms have their oxidation state changed. ** Organic redox reaction, a redox reacti ...

'' of ''A'', i.e. its orthogonal complement is also invariant under ''A''.

References

*P. Halmos, A Hilbert Space Problem Book, Springer, New York 1982. Operator theory Invariant subspaces Linear operators