Singular values of a compact operator - Imported Knowlpedia Content

Definition

Let $T:H\to K$ be a compact operator between Hilbert spaces. Its absolute value is the positive compact operator

|T|=(T^*T)^{1/2}\quad\text{on }H.

The singular values $s_1(T)\geq s_2(T)\geq\cdots\geq0$ are the nonzero eigenvalues of $|T|$ , repeated according to multiplicity and arranged in nonincreasing order, followed by zeros when appropriate. Equivalently, for $n\geq1$ ,

s_n(T)=\inf\{\|T-F\|:\operatorname{rank}(F)<n\}.

This approximation-number formulation also fixes the zero tail and remains unambiguous when $H$ or $K$ is not separable.

Fundamental properties

The first singular value is the operator norm: $s_1(T)=\|T\|$ . Compactness is equivalent to $s_n(T)\to0$ when the zero tail is included, and $s_n(T)=s_n(T^*)$ for every $n$ . Unitary changes of coordinates on the source or target do not alter the sequence. These facts make singular values the coordinate-free infinite-dimensional analogue of the singular values of a matrix Simon, §§1–2.

Approximation and ideal estimates

The formula using rank- $<n$ operators measures the best possible operator-norm approximation of $T$ by low-rank maps. It implies, for bounded operators $A$ and $B$ ,

s_n(ATB)\leq\|A\|\,s_n(T)\,\|B\|.

Hence decay conditions on the sequence define two-sided operator ideals. Summability of $(s_n(T))$ gives the trace-class and Schatten conditions, while weaker rates lead to weak Schatten ideals.

Examples and boundary cases

For the diagonal operator $T(x_1,x_2,\ldots)=(\lambda_1x_1,\lambda_2x_2,\ldots)$ on $\ell^2$ , with $\lambda_n\to0$ , the singular values are the numbers $|\lambda_n|$ rearranged in decreasing order. A rank- $r$ operator has $s_n(T)=0$ for $n>r$ . A bounded diagonal operator whose diagonal entries do not tend to zero is not compact, so it does not have a singular-value sequence tending to zero in this sense.

References

Barry Simon, Trace Ideals and Their Applications, 2nd ed., American Mathematical Society, 2005. DOI record. Relevant: §§1–2 on singular values, approximation numbers, and ideal inequalities.
Israel Gohberg and Mark Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, American Mathematical Society, 1969. AMS record. Relevant: Chapter II on characteristic numbers of compact operators.