Weyl product and sum inequalities for compact operators

By [A1, A3, A4], the nonzero spectral values are eigenvalues of finite algebraic multiplicity, and there are only finitely many above any positive modulus threshold. Every nonzero spectral value lies in the threshold set $\{\lambda :\mathrm{\mid }\lambda \mathrm{\mid }\ge 1\mathrm{/}m\}$ for some $m\ge 1$ by [A23, A24]. Each threshold set is finite by [A3], so Countable Choice [A2] and [A12] make their union at most countable. Repeating each value according to its finite algebraic multiplicity still gives a countable list by another application of [A12]. If that list is infinite, after choosing any remaining value only finitely many remaining values have at least its modulus; that finite nonempty set has a maximum modulus by [A22], which is then maximal among all remaining values. Finite ties can be ordered arbitrarily. AC permits iterating this selection to obtain a sequence ordered by decreasing modulus. If $N=0$, both products are the empty product $1$. If $N\ge 1$ and the zero-padded eigenvalue list has ${\lambda }_N(T)=0$, its first $N$-term product is zero and the asserted inequality follows from nonnegativity of singular values. It remains to prove the claim when $N\ge 1$ and all of ${\lambda }_1(T),\dots ,{\lambda }_N(T)$ are nonzero.

For each distinct eigenvalue $\lambda$ among this prefix, let $r_{\lambda }$ be its number of occurrences. Then $r_{\lambda }\le \dim G_{\lambda }(T)$ by [A4]. Apply [A5] to $(T-\lambda I){\mathrm{\mid }}_{G_{\lambda }(T)}$ and list the resulting finite Jordan-string lengths as ${\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_q$. In that order, take an initial segment of length $k_i:=\min ({\mathrm{\ell }}_i,r_{\lambda }-{\sum }_{h<i}{k}_h)$ from string $i$ while the residual is positive, and take length zero thereafter. Since ${\sum }_i{\mathrm{\ell }}_i=\dim G_{\lambda }(T)\ge r_{\lambda }$, these lengths sum to $r_{\lambda }$. The definition in [A5] makes each selected initial segment invariant under $T$, so their direct sum $F_{\lambda }$ is $T$-invariant, has dimension $r_{\lambda }$, and the restriction of $T$ to it is triangular with diagonal entry $\lambda$ repeated $r_{\lambda }$ times.

Put $S:={\mathrm{\Lambda }}^{N}T$, $K:=(\ker T{)}^{\perp }$, and $P:=U^{\ast }U$, the orthogonal projection onto $K$ from [A6]. For every increasing $N$-tuple $J=(j_1<\cdots <j_N)$ in $J_T$, set ${\eta }_J:=e_{j_1}\wedge \cdots \wedge e_{j_N}$, ${\theta }_J:=f_{j_1}\wedge \cdots \wedge f_{j_N}$, and ${\mu }_J:={\prod }_{k=1}^N{s}_{j_k}(T)$. By the Gram formula [A7], both wedge families are orthonormal. The $e_j$ span $K$: if their closed span were proper, [A11] would give a nonzero $x\in K$ orthogonal to all $e_j$, while the SVD expansion [A6] would imply $Tx=0$, contradicting $K\cap \ker T=\{0\}$. Since decomposable wedges are dense by the exterior construction, approximating each factor by finite $e_j$-sums and expanding by multilinearity shows that the ${\eta }_J$ span ${\mathrm{\Lambda }}^{N}K$. The tensor projection $P^{\otimes N}$ is self-adjoint and idempotent and commutes with permutations; its restriction to the alternating range is ${\mathrm{\Lambda }}^{N}P$, the orthogonal projection onto $M:=\stackrel{}{\mathrm{span}}\{{\eta }_J\}$. Since $T=TP$, functoriality [A7] gives $S=S{\mathrm{\Lambda }}^{N}P$, so $S$ vanishes on $M^{\perp }$. On each ${\eta }_J$, $S{\eta }_J={\mu }_J{\theta }_J$. If there are at least $N$ positive singular values, the family $({\eta }_J)$ is a complete orthonormal family in $M$. For $x\in M$, Parseval and net convergence [A13] give $x_F:={\sum }_{J\in F}{c}_J{\eta }_J\to x$ over finite subsets $F$, with $c_J=⟨x,{\eta }_J⟩$. Orthonormality of both wedge families gives $\mathrm{\parallel }S{x}_F{\mathrm{\parallel }}^2={\sum }_{J\in F}{\mu }_J^2\mathrm{\mid }{c}_J{\mathrm{\mid }}^2\le ({\sup }_J{\mu }_J{)}^2{\sum }_{J\in F}\mathrm{\mid }{c}_J{\mathrm{\mid }}^2=({\sup }_J{\mu }_J{)}^2\mathrm{\parallel }{x}_F{\mathrm{\parallel }}^2$. Since $S$ is bounded [A7], passing to the norm limit yields $\mathrm{\parallel }Sx\mathrm{\parallel }\le ({\sup }_J{\mu }_J)\mathrm{\parallel }x\mathrm{\parallel }$. The unit vector ${\eta }_{(1,\dots ,N)}$ attains ${\sup }_J{\mu }_J={\prod }_{j=1}^N{s}_j(T)$, because every increasing tuple has $s_{j_k}(T)\le s_k(T)$. If there are fewer than $N$ positive singular values, $K$ is finite-dimensional of dimension less than $N$, so every decomposable $N$-wedge in $K$ vanishes by [A14]; density gives ${\mathrm{\Lambda }}^{N}K=\{0\}$ and $S=0$, the same norm formula holding with $s_N(T)=0$.

The spaces $G_{\lambda }(T)$ for the finitely many distinct prefix values form a direct sum: if ${\sum }_{\nu }{x}_{\nu }=0$ with $x_{\nu }\in G_{\nu }(T)$, fix $\lambda$ and apply $Q_{\lambda }:={\prod }_{\nu \mathrm{\ne }\lambda }(T-\nu I{)}^{m_{\nu }}$. It kills every $x_{\nu }$ for $\nu \mathrm{\ne }\lambda$. On $G_{\lambda }(T)$ each factor is $(\lambda -\nu )I+N_{\lambda }$, where $N_{\lambda }=(T-\lambda I){\mathrm{\mid }}_{G_{\lambda }}$ is nilpotent, so that factor is invertible by its finite geometric-series inverse. Thus $Q_{\lambda }{\mathrm{\mid }}_{G_{\lambda }}$ is invertible and $x_{\lambda }=0$. Consequently $E_N:={⨁}_{\lambda }{F}_{\lambda }$ is a $T$-invariant $N$-dimensional subspace.

Choose the concatenated Jordan-segment basis $v_1,\dots ,v_N$ of $E_N$ and put $w:=v_1\wedge \cdots \wedge v_N$. The vectors are linearly independent by their being a basis, so [A14] gives $w\mathrm{\ne }0$. If $A$ is the triangular matrix of $T{\mathrm{\mid }}_{E_N}$ in this basis, multilinearity and alternation [A7] expand $(T{v}_1)\wedge \cdots \wedge (T{v}_N)=\det (A)w$. By [A8], $\det (A)={\prod }_{j=1}^N{\lambda }_j(T)$. Hence $({\mathrm{\Lambda }}^{N}T)w=({\prod }_{j=1}^N{\lambda }_j(T))w$.

In the nonzero-prefix case of step 1.1, step 3.1 gives an eigenvector of $S$ with eigenvalue ${\prod }_{j=1}^N{\lambda }_j(T)$. The operator-norm bound [A9] and norm identity [step 1.3] yield ${\prod }_{j=1}^N\mathrm{\mid }{\lambda }_j(T)\mathrm{\mid }\le \mathrm{\parallel }S\mathrm{\parallel }={\prod }_{j=1}^N{s}_j(T)$. Together with step 1.1 this proves the product inequality for every $N\ge 0$.

Fix $N\ge 1$ with ${\lambda }_1(T),\dots ,{\lambda }_N(T)$ all nonzero. For each $k\le N$, step 4.1 gives the product inequality for the first $k$ terms. Its left side is positive, hence $s_k(T)>0$. The eigenvalue moduli are positive by [A24], so set $x_j:=\log \mathrm{\mid }{\lambda }_j(T)\mathrm{\mid }$ and $y_j:=\log s_j(T)$ for $j\le N$. Both sequences are nonincreasing because modulus, singular values and logarithm preserve the indicated order; taking logarithms in the product inequalities and using the logarithm product law [A20] gives ${\sum }_{j=1}^k{x}_j\le {\sum }_{j=1}^k{y}_j$ for every $k\le N$.

For a nonincreasing real list $x_1,\dots ,x_N$ and real $t$, ${\sum }_{j=1}^N(x_j-t{)}_{+}={\max }_{0\le k\le N}({\sum }_{j=1}^k{x}_j-kt)$, where the $k=0$ sum is zero: the positive terms form an initial segment, and adding any nonpositive later term cannot increase the prefix sum. This finite maximum exists by [A22]. Applying the identity to the lists in step 5.1 and their prefix-sum inequalities gives ${\sum }_{j=1}^N(x_j-t{)}_{+}\le {\sum }_{j=1}^N(y_j-t{)}_{+}$ for every $t\in \mathbb{R}$. Put $c:=\min \{x_N,y_N\}-1$ and $b:=\max \{x_1,y_1\}+1$, so $c<x_j,y_j<b$ for every $j$, and for $R\ge 0$ put $a_R:=c-R$ and ${\varphi }_u(t):=(u-t{)}_{+}\exp (t)$. By [A15, A16], these functions and their finite sums are continuous and integrable on $[a_R,b]$; since $\exp (t)>0$, the positive-part inequality remains true after multiplication by $\exp (t)$. Monotonicity and linearity of the integral [A16] give ${\sum }_j{\int }_{a_R}^b{\varphi }_{x_j}(t)dt\le {\sum }_j{\int }_{a_R}^b{\varphi }_{y_j}(t)dt$. Fix any $u$ among these finitely many $x_j,y_j$. On $[a_R,u]$, ${\varphi }_u(t)=(u-t)\exp (t)$, whose primitive is $G_u(t):=(u-t+1)\exp (t)$: directly from the derivative definition [A17], $(u-t+1{)}^{\mathrm{\prime }}=-1$, and the product rule and $(\exp {)}^{\mathrm{\prime }}=\exp$ give $G_u^{\mathrm{\prime }}(t)=(u-t)\exp (t)$. On $[u,b]$, ${\varphi }_u=0$. Applying the fundamental theorem, the zero-integral case and interval additivity [A16, A17] yields ${\int }_{a_R}^b{\varphi }_u(t)dt=\exp (u)-(u-a_R+1)\exp (a_R)$. The error $E_{u,R}:=(u-a_R+1)\exp (a_R)=(R+u-c+1)\exp (c-R)$ is nonnegative and tends to zero: for sufficiently large $R$, it is at most $2\exp (c)R\mathrm{/}\exp (R)$, which tends to zero by [A18]. Because there are finitely many $x_j$, for every $\epsilon >0$ one common sufficiently large $R$ makes ${\sum }_j{E}_{x_j,R}<\epsilon$. The integral inequality then gives ${\sum }_j\exp (x_j)\le {\sum }_j\exp (y_j)+\epsilon$, since $E_{y_j,R}\ge 0$. As $\epsilon$ was arbitrary, [A19] gives ${\sum }_{j=1}^N\mathrm{\mid }{\lambda }_j(T)\mathrm{\mid }\le {\sum }_{j=1}^N{s}_j(T)$.

If the nonzero eigenvalue list is finite with length $m>0$, step 6.1 at $N=m$ bounds its full absolute sum; for $m=0$ that sum is zero. If the list is infinite, step 6.1 bounds every eigenvalue partial sum by the corresponding singular-value partial sum, which is at most $\mathrm{\parallel }T{\mathrm{\parallel }}_1$ by trace class [A10]. The eigenvalue terms are nonnegative, so [A21] gives convergence of their series and bounds its sum, the supremum of its partial sums, by $\mathrm{\parallel }T{\mathrm{\parallel }}_1$. Zero padding changes neither sum, and [A10] identifies the singular-value sum with $\mathrm{\parallel }T{\mathrm{\parallel }}_1$. This proves the sum conclusion, including finite-rank and zero operators.