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In a unital Banach algebra, for $\lambda \ne 0$ the existence of $R=(\lambda 1-ab{)}^{-1}$ gives $(\lambda 1-ba{)}^{-1}={\lambda }^{-1}(1+bRa)$, as verified by multiplying on either side. Interchanging $a,b$ proves the converse. Zero may differ: for the unilateral shift operator $S$ on ${\ell }^2({\mathbb{N}}_0)$, $S^{\ast }S=1$ has spectrum $\{1\}$ and $S{S}^{\ast }=1-P_0$ has spectrum $\{0,1\}$.

An element of a complex unital Banach algebra is quasinilpotent if its spectral radius is zero, equivalently its spectrum of an element is $\{0\}$. A nilpotent element is quasinilpotent, but the converse need not hold. The Volterra convolution algebra on a finite interval gives powers which never vanish, yet have factorially decreasing norms, making the inverse series for $\lambda e-x$ converge for every $\lambda \ne 0$.

The identity $(x+\alpha e)-\lambda e=x-(\lambda -\alpha )e$ proves the formula directly from invertibility. In a nonzero complex unital Banach algebra, the nonemptiness of the Banach-algebra spectrum implies that if $x$ is a quasinilpotent element, then $\rho (x+e)=1$. Consequently every nonscalar element cannot simultaneously be quasinilpotent: adding the identity preserves nonscalarity while translating its spectrum.

For a closed unital subalgebra $B$ of a unital Banach algebra $A$ and $x\in B$, the smaller-algebra algebra resolvent set is relatively open and closed in the larger one. Openness follows from the Neumann series; closedness follows from continuity of inversion in $A$ and norm closedness of $B$. Every larger-algebra resolvent component is therefore either retained whole or absorbed into the smaller-algebra Banach algebra spectrum. The unbounded component is always retained, since inverses at large modulus are norm-convergent series in $x$.

The algebra resolvent set consists of those scalars for which $\lambda 1-x$ is invertible in the algebra. It is open by the Neumann series, and the inverse-valued map is the resolvent of an element. A closed subalgebra can have a smaller algebra resolvent set because an ambient inverse need not lie in it.

Spectrophotometry measures received light as a function of wavelength, accounting for the instrument's wavelength-dependent response.

The Biot strain tensor uses $f(\lambda )=\lambda -1$, so $E^{(1)}=U-I$. Its principal values are extension ratios minus one. It is a spectral strain measure, since $f(1)=0$, $f^{\prime }(1)=1$ and $f^{\prime }>0$. Its reference-volume work conjugate is the symmetric Biot stress.

In vacuum, $\nu =c/\lambda$, where $c$ is the speed of light. Increasing wavelength decreases frequency, so the ordinary signed differentials satisfy $d\nu =-c{\lambda }^{-2}d\lambda$. A spectral slice's positive contribution to radiance must be unchanged. The change of variables formula therefore gives the absolute Jacobian determinantThese formulas use $L_{\lambda }$ per metre and $L_{\nu }$ per hertz. The factor $c/{\nu }^2$ has units $\mathrm{m}\mathrm{s}=\mathrm{m}\mathrm{H}{\mathrm{z}}^{-1}$, which converts the former density into the latter. If ${\ell }_{\mathrm{nm}}$ denotes the numerical density per nanometre in $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}\mathrm{n}{\mathrm{m}}^{-1}$, its per-metre value is $10^9{\ell }_{\mathrm{nm}}$. Conversely, the per-nanometre value is $10^{-9}$ times the per-metre value. Likewise, a density per terahertz is $10^{12}$ times its per-hertz value because $1\mathrm{THz}=10^{12}\mathrm{Hz}$.

For a concrete stipulated spectrum, let $L_{\lambda }=C=10^9\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}{\mathrm{m}}^{-1}$ between ${\lambda }_1=500\mathrm{nm}$ and ${\lambda }_2=600\mathrm{nm}$, and zero outside. This is exactly $1\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}\mathrm{n}{\mathrm{m}}^{-1}$ throughout the band. Using the exact International System of Units value $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$, the corresponding frequency bounds areThe transformed spectral radiance is $L_{\nu }=Cc/{\nu }^2$ on this interval. At ${\nu }_1$ it is approximately $1.200830743\times 10^{-12}\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}\mathrm{H}{\mathrm{z}}^{-1}$, and at ${\nu }_2$ it is $8.339102380\times 10^{-13}$ in the same units. These are respectively $1.200830743$ and $0.833910238$ per terahertz. Thus a flat wavelength density becomes a decreasing frequency density; relabelling the horizontal axis alone would miss both the shape and the units. ^[ref].

The equality of band radiances has a short exact proof:The reversed bounds and the Jacobian determinant preserve the same spectral measure. Each half-band, $500$–$550\mathrm{nm}$ and $550$–$600\mathrm{nm}$, carries $50\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$, although the corresponding frequency widths are about $54.508$ and $45.423\mathrm{THz}$.

A spectrum's peak is consequently a statement about a particular density and coordinate. The frequency and wavelength peaks of Planck's law differ for precisely this reason. Integrated radiance over a specified physical band supplies a comparison that survives the change of variables.

A Voigt profile is the convolution of a Gaussian distribution and a Cauchy distribution. For an absorption line, thermal Doppler broadening contributes the Gaussian core while finite excited-state lifetime contributes the Lorentzian wings. Instrumental broadening and unresolved motions must also be modeled when inferring column density and temperature.

Collisions and fluctuating neighboring-particle electric fields alter atomic energy levels and coherence times, broadening a spectral line. In a simple impact treatment the coherence-decay rate gives a Lorentzian width of order $\Gamma /(2\pi )$; the precise definition of full or half width must be stated. Unlike Doppler broadening, this depends primarily on collision rates and perturber density. In stellar opacity calculations it alters line overlap and the low-opacity windows entering the Rosseland mean opacity.

For a nonnegative elliptic operator $L$ on a closed manifold, sum its positive eigenvalues with multiplicity. The defining series converges in a right half-plane. If $n_0=\dim \ker L$ and $K(t)=\mathrm{Tr}(e^{-tL})$ is the heat trace, thenSubtracting finitely many terms of the short-time heat kernel expansion gives a meromorphic continuation. Zero eigenvalues must be excluded: assigning them a nonexistent inverse power does not define the spectral zeta function.

For the positive scalar Laplace-Beltrami operator on a closed $d$-dimensional Riemannian manifold, the number of eigenvalues at most $\lambda$, counted with multiplicity, has the displayed leading asymptotic. The exponent determines the dimension and the coefficient determines the total volume of the Riemannian metric. There is an analogous result with fixed Dirichlet boundary conditions or Neumann boundary conditions on a compact smooth domain.

For the positive Laplace-Beltrami operator $P$ on a closed manifold, the wave trace is the distribution ${\sum }_j{e}^{it\sqrt{{\lambda }_j}}$, counting multiplicities. The series is interpreted after testing against smooth compactly supported functions, rather than as an ordinarily convergent sum. Its singularities relate the spectrum to the length spectrum of closed geodesics, through the Duistermaat-Guillemin trace formula.

In units $c=1$, an isentropic perfect fluid in general relativity has sound speed $c_s^2=dp/d\rho$. Writing $n=p^{\prime }(ϵ)$ gives $d\rho =ϵdn$ and $c_s^2=n/(ϵp^{\prime \prime }(ϵ))$. Linearized particle number conservation in a relativistic fluid and the relativistic Euler equation then yield acoustic waves with this speed. Positive $c_s^2$ is needed for ordinary acoustic stability; a causal fluid model additionally requires $c_s^2\le 1$.

For an integer $s\ge 1$, the last-column map $p(A)=A{e}_{s+1}$ has fiber $SO(s)$, included as $\mathrm{diag}(B,1)$. Local completion of a unit vector by the Gram-Schmidt process gives local fiber bundle charts. Consequently it has the homotopy lifting property and the long exact sequence of homotopy groups of a fibration.

All the special orthogonal groups are connected, and their second homotopy groups vanish. The rotation circle gives $SO(2)\cong S^1$. Conjugation by unit quaternions gives the twofold universal cover $S^3\to SO(3)$, so ${\pi }_{1}SO(3)=\mathbb{Z}/2$ and ${\pi }_{3}SO(3)=\mathbb{Z}$. The special orthogonal sphere fibration $SO(n-1)\to SO(n)\to S^{n-1}$ propagates the first two homotopy groups for $n\ge 4$. In dimension four its exact sequence has a zero map from the torsion group ${\pi }_4{S}^3=\mathbb{Z}/2$ to ${\pi }_{3}SO(3)=\mathbb{Z}$, giving ${\pi }_{3}SO(4)={\mathbb{Z}}^2$. For $SO(1)$ all positive groups vanish, and ${\pi }_{3}SO(2)=0$.

For positive distinct diagonal entries $c_1<\cdots <c_n$, the critical points of this function are diagonal sign matrices $D=\mathrm{diag}({\epsilon }_1,\dots ,{\epsilon }_n)$ with $\prod {\epsilon }_i=1$. Criticality forces $CX$ symmetric; its square is $C^2$, so it commutes with the distinct-spectrum diagonal matrix $C^2$ and is diagonal. In the skew coordinates $a_{ij}$ its Hessian is $-{\sum }_{i<j}(c_i{\epsilon }_i+c_j{\epsilon }_j)a_{ij}^2$. The sign of each parenthesis is that of ${\epsilon }_j$, giving Morse index ${\sum }_{j:{\epsilon }_j=1}(j-1)$.

The lightest supersymmetric particle is the least massive sparticle in a given spectrum. Exact R-parity makes it stable: a lighter final state cannot contain another odd particle, while a final state of ordinary particles is even. A neutral weakly interacting lightest supersymmetric particle is a possible dark matter constituent, but its cosmological abundance must be computed separately.

In the MSSM, electroweak symmetry breaking mixes the neutral bino, wino and two neutral higgsinos into four neutral Majorana spinor mass eigenstates, the neutralinos. A neutralino can be the lightest supersymmetric particle; with conserved R-parity it then provides a stable possible dark matter candidate.

A maximal ideal $M$ of $C(\mathbb{T})$ has a common zero: otherwise finitely many $f_j\in M$ have no simultaneous zero, and ${\sum }_j|f_j{|}^2\in M$ is invertible, a contradiction. If the common zero is $z$, then $M$ is the kernel of evaluation at $z$ by maximality. Thus every algebra character is point evaluation. Continuous functions separate distinct points of the unit circle, and the evaluation bijection is a homeomorphism for the Gelfand topology.